A Survey on Migration-Selection Models
in Population Genetics1
arXiv:1309.2576v1 [q-bio.PE] 10 Sep 2013
Reinhard Bürger
Department of Mathematics, University of Vienna
Contact:
Reinhard Bürger
Institut für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
A-1090 Wien
Austria
E-mail: [email protected]
Phone: +43 1 4277 50631
Fax: +43 1 4277 9506
Key words and phrases: differential equations, recurrence equations, stability, convergence, perturbation theory, evolution, geographic structure, dispersal, recombination
2010 Mathematics Subject Classification: 92D10, 92D15; 34D05, 34D30, 37N25, 39A60
Running head: Migration-selection models
1
Acknowledgments. This survey is based on lecture notes I compiled for a lectures series presented
during the “Program on Nonlinear Equations in Population Biology” in June 2013 at the Center for PDE,
East China Normal University, Shanghai. In turn, these lecture notes were a revised version of notes
written for a lecture course at the University of Vienna in 2010. I am very grateful to the participants
of these courses for many helpful comments, in particular to Dr. Simon Aeschbacher, Benja Fallenstein,
and Dr. Linlin Su. Comments by and useful discussions with Professor Yuan Lou helped to finalize this
work. I am particularly grateful to Professor Wei-Ming Ni for inviting me to the Center for PDE, and
to him and its members for their great hospitality. Financial support by grant P25188 of the Austrian
Science Fund FWF is gratefully acknowledged.
1
Abstract
This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models
are most appropriate if the dispersal distance is short compared to the scale at which the
environment changes, or if the habitat is fragmented. The general goal of such models
is to study the influence of population subdivision and gene flow among subpopulations
on the amount and pattern of genetic variation maintained. Only deterministic models are treated. Because space is discrete, they are formulated in terms of systems of
nonlinear difference or differential equations. A central topic is the exploration of the
equilibrium and stability structure under various assumptions on the patterns of selection
and migration. Another important, closely related topic concerns conditions (necessary
or sufficient) for fully polymorphic (internal) equilibria. First, the theory of one-locus
models with two or multiple alleles is laid out. Then, mostly very recent, developments
about multilocus models are presented. Finally, as an application, analysis and results of
an explicit two-locus model emerging from speciation theory are highlighted.
Contents
1 Introduction
4
2 Selection on a multiallelic locus
2.1 The Hardy–Weinberg Law . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
2.2
Evolutionary dynamics under selection . . . . . . . . . . . . . . . . . . . .
2.3
Two alleles and the role of dominance . . . . . . . . . . . . . . . . . . . . . 12
2.4
The continuous-time selection model . . . . . . . . . . . . . . . . . . . . . 14
3 The general migration-selection model
8
16
3.1
The recurrence equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2
The relation between forward and backward migration rates . . . . . . . . 18
3.3
Important special migration patterns . . . . . . . . . . . . . . . . . . . . . 20
4 Two alleles
22
4.1
Protected polymorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2
Two demes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2
4.3
Arbitrary number of demes . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4
4.5
The continent-island model . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Submultiplicative fitnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 The Levene model
39
5.1
General results about equilibria and dynamics . . . . . . . . . . . . . . . . 40
5.2
5.3
No dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Two alleles with dominance . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Multiple alleles and arbitrary migration
50
6.1
No dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2
6.3
Migration and selection in continuous time . . . . . . . . . . . . . . . . . . 51
Weak migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.4
Strong migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.5
Uniform selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 Multilocus models
60
7.1 Recombination between two loci . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2
Two diallelic loci under selection
7.3
The general model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.4
7.5
Migration and recombination . . . . . . . . . . . . . . . . . . . . . . . . . 68
Weak selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.6
Strong migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.7
Weak recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.8
7.9
Weak migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Weak evolutionary forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.10 The Levene model
. . . . . . . . . . . . . . . . . . . . . . . 64
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.11 Some conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8 Application: A two-locus model for the evolution of genetic incompatibilities with gene flow
82
8.1
The haploid DMI model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2
Existence and stability of boundary equilibria . . . . . . . . . . . . . . . . 84
8.3
8.4
Properties of internal equilibria . . . . . . . . . . . . . . . . . . . . . . . . 86
Bifurcations of two internal equilibria . . . . . . . . . . . . . . . . . . . . . 89
8.5
No migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3
8.6
Weak migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.7
The complete equilibrium and stability structure . . . . . . . . . . . . . . . 92
9 References
1
98
Introduction
Population genetics is concerned with the study of the genetic composition of populations.
This composition is shaped by selection, mutation, recombination, mating behavior and
reproduction, migration, and other genetic, ecological, and evolutionary factors. Therefore, these mechanisms and their interactions and evolutionary consequences are investigated. Traditionally, population genetics has been applied to animal and plant breeding,
to human genetics, and more recently to ecology and conservation biology. One of the
main subjects is the investigation of the mechanisms that generate and maintain genetic
variability in populations, and the study of how this genetic variation, shaped by environmental influences, leads to evolutionary change, adaptation, and speciation. Therefore,
population genetics provides the basis for understanding the evolutionary processes that
have led to the diversity of life we encounter and admire.
Mathematical models and methods have a long history in population genetics, tracing
back to Gregor Mendel, who used his education in mathematics and physics to draw his
conclusions. Francis Galton and the biometricians, notably Karl Pearson, developed new
statistical methods to describe the distribution of trait values in populations and to predict their change between generations. Yule (1902), Hardy (1908), and Weinberg (1908)
worked out simple, but important, consequences of the particulate mode of inheritance
proposed by Mendel in 1866 that contrasted and challenged the then prevailing blending
theory of inheritance. However, it was not before 1918 that the synthesis between genetics and the theory of evolution through natural selection began to take shape through
Fisher’s (1918) work. By the early 1930s, the foundations of modern population genetics
had been laid by the work of Ronald A. Fisher, J.B.S. Haldane, and Sewall Wright. They
had demonstrated that the theory of evolution by natural selection, proposed by Charles
Darwin in 1859, can be justified on the basis of genetics as governed by Mendel’s laws. A
detailed account of the history of population genetics is given in Provine (1971).
In the following, we explain some basic facts and mechanisms that are needed throughout this survey. Mendel’s prime achievement was the recognition of the particulate nature
4
of the hereditary determinants, now called genes. Its position along the DNA is called
the locus, and a particular sequence there is called an allele. In most higher organisms,
genes are present in pairs, one being inherited from the mother, the other from the father.
Such organisms are called diploid. The allelic composition is called the genotype, and the
set of observable properties derived from the genotype is the phenotype.
Meiosis is the process of formation of reproductive cells, or gametes (in animals, sperm
and eggs) from somatic cells. Under Mendelian segregation, each gamete contains precisely
one of the two alleles of the diploid somatic cell and each gamete is equally likely to contain
either one. The separation of the paired alleles from one another and their distribution to
the gametes is called segregation and occurs during meiosis. At mating, two reproductive
cells fuse and form a zygote (fertilized egg), which contains the full (diploid) genetic
information.
Any heritable change in the genetic material is called a mutation. Mutations are
the ultimate source of genetic variability and form the raw material upon which selection
acts. Although the term mutation includes changes in chromosome structure and number,
the vast majority of genetic variation is caused by changes in the DNA sequence. Such
mutations occur in many different ways, for instance as base substitutions, in which one
nucleotide is replaced by another, as insertions or deletions of DNA, as inversions of
sequences of nucleotides, or as transpositions. For the population-genetic models treated
in this text the molecular origin of a mutant is of no relevance because they assume that
the relevant alleles are initially present.
During meiosis, different chromosomes assort independently and crossing over between
two homologous chromosomes may occur. Consequently, the newly formed gamete contains maternal alleles at one set of loci and paternal alleles at the complementary set.
This process is called recombination. Since it leads to random association between alleles
at different loci, recombination has the potential to combine favorable alleles of different
ancestry in one gamete and to break up combinations of deleterious alleles. These properties are generally considered to confer a substantial evolutionary advantage to sexual
species relative to asexuals.
The mating pattern may have a substantial influence on the evolution of gene frequencies. The simplest and most important mode is random mating. This means that matings
take place without regard to ancestry or the genotype under consideration. It seems to
occur frequently in nature. For example, among humans, matings within a population
appear to be random with respect to blood groups and allozyme phenotypes, but are
nonrandom with respect to other traits, for example, height.
5
Selection occurs when individuals of different genotype leave different numbers of
progeny because they differ in their probability to survive to reproductive age (viability), in their mating success, or in their average number of produced offspring (fertility).
Darwin recognized and documented the central importance of selection as the driving force
for adaptation and evolution. Since selection affects the entire genome, its consequences
for the genetic composition of a population may be complex. Selection is measured in
terms of fitness of individuals, i.e., by the number of progeny contributed to the next
generation. There are different measures of fitness, and it consists of several components
because selection may act on each stage of the life cycle.
Because many natural populations are geographically structured and selection varies
spatially due to heterogeneity in the environment, it is important to study the consequences of spatial structure for the evolution of populations. Dispersal of individuals is
usually modeled in one of two alternative ways, either by diffusion in space or by migration between discrete niches, or demes. If the population size is sufficiently large, so
that random genetic drift can be ignored, then the first kind of model leads to partial
differential equations (Fisher 1937, Kolmogorov et al. 1937). This is a natural choice if
genotype frequencies change continuously along an environmental gradient, as it occurs
in a cline (Haldane 1948). Here we will not be concerned with this wide and fruitful area
and refer instead to Barton (1999), Nagylaki and Lou (2008), Barton and Turelli (2011),
Lou et al. (2013) for recent developments and references.
Instead, this survey focuses on models of selection and migration between discrete
demes. Such models are most appropriate if the dispersal distance is short compared
to the scale at which the environment changes, or if the habitat is fragmented. They
originated from the work of Haldane (1930) and Wright (1931). Most of the existing
theory has been devoted to study selection on a single locus in populations with discrete,
nonoverlapping generations that mate randomly within demes. However, advances in
the theory of multilocus models have been made recently. The general goal is to study
the influence of population subdivision and of gene flow among subpopulations on the
amount and pattern of genetic variation maintained. The models are formulated in terms
of systems of nonlinear difference or differential equations. The core purpose of this
survey is the presentation of the methods of their analysis, of the main results, and of
their biological implications.
For mathematically oriented introductions to the much broader field of population
genetics, we refer to the books of Nagylaki (1992), Bürger (2000), Ewens (2004), and
Wakeley (2008). The two latter texts treat stochastic models in detail, an important and
6
topical area ignored in this survey.
2
Selection on a multiallelic locus
Darwinian evolution is based on selection and inheritance. In this section, we summarize
the essential properties of simple selection models. It will prepare the ground for the
subsequent study of the joint action of spatially varying selection and migration. Proofs
and a detailed treatment may be found in Chapter I of Bürger (2000). Our focus is on
the evolution of the genetic composition of the population, but not on its size. Therefore,
we always deal with relative frequencies of genes or genotypes within a given population.
Unless stated otherwise, we consider a population with discrete, nonoverlapping generations, such as annual plants or insects. We assume two sexes that need not be distinguished because gene or genotype frequencies are the same in both sexes (as is always the
case in monoecious species). Individuals mate at random with respect to the locus under
consideration, i.e., in proportion to their frequency. We also suppose that the population is large enough that gene and genotype frequencies can be treated as deterministic,
and relative frequency can be identified with probability. Then the evolution of gene or
genotype frequencies can be described by difference or recurrence equations. These assumptions reflect an idealized situation which will model evolution at many loci in many
populations or species, but which is by no means universal.
2.1
The Hardy–Weinberg Law
With the blending theory of inheritance variation in a population declines rapidly, and
this was one of the arguments against Darwin’s theory of evolution. With Mendelian
inheritance there is no such dilution of variation, as was shown independently by the famous British mathematician Hardy (1908) and, in much greater generality, by the German
physician Weinberg (1908, 1909).
We consider a single locus with I possible alleles Ai and write I = {1, . . . , I} for the set
of all alleles. We denote the frequency of the ordered genotype Ai Aj by Pij , so that the
frequency of the unordered genotype Ai Aj is Pij + Pji = 2Pij . Subscripts i and j always
7
refer to alleles. Then the frequency of allele Ai in the population is
pi =
I
X
Pij .2
j=1
After one generation of random mating the zygotic proportions satisfy3
Pij0 = pi pj
for every i and j .
A mathematically trivial, but biologically very important, consequence is that (in the
absence of other forces) gene frequencies remain constant across generations, i.e.,
p0i = pi
for every i .
(2.1)
In other words, in a (sufficiently large) randomly mating population reproduction does
not change allele frequencies. A population is said to be in Hardy–Weinberg equilibrium
if
Pij = pi pj .
(2.2)
In a (sufficiently large) randomly mating population, this relation is always satisfied among
zygotes.
Evolutionary mechanisms such as selection, migration, mutation, or random genetic
drift distort Hardy-Weinberg proportions, but Mendelian inheritance restores them if
mating is random.
2.2
Evolutionary dynamics under selection
Selection occurs when genotypes in a population differ in their fitnesses, i.e., in their
viability, mating success, or fertility and, therefore, leave different numbers of progeny.
The basic mathematical models of selection were developed and investigated in the 1920s
and early 1930s by Fisher (1930), Wright (1931), and Haldane (1932).
We will be concerned with the evolutionary consequences of selection caused by differential viabilities, which leads to simpler models than (general) fertility selection (e.g.,
Hofbauer and Sigmund 1988, Nagylaki 1992). Suppose that at an autosomal locus the alleles A1 , . . . , AI occur. We count individuals at the zygote stage and denote the (relative)
frequency of the ordered genotype Ai Aj by Pij (= Pji ). Since mating is at random, the
2
P
P
If no summation range is indicated, it is assumed to be over all admissible values; e.g., i = i∈I
3
Unless stated otherwise, a prime, 0 , always signifies the next generation. Thus, instead of Pij (t) and
Pij (t + 1), we write Pij and Pij0 (and analogously for other quantities).
8
genotype frequencies Pij are in Hardy-Weinberg proportions. We assume that the fitness
(viability) wij of an Ai Aj individual is constant, i.e., independent of time, population
size, or genotype frequencies. In addition, we suppose wij = wji , as is usually the case.
Then the frequency of Ai Aj genotypes among adults that have survived selection is
Pij∗ =
wij Pij
wij pi pj
=
,
w̄
w̄
where we have used (2.2). Here,
X
X
X
w̄ =
wij Pij =
wij pi pj =
w i pi
i,j
i,j
(2.3)
i
is the mean fitness of the population and
wi =
X
wij pj
(2.4)
j
is the marginal fitness of allele Ai . Both are functions of p = (p1 , . . . , pI )> .4
Therefore, the frequency of Ai after selection is
X
wi
p∗i =
Pij∗ = pi .
w̄
j
(2.5)
Because of random mating, the allele frequency p0i among zygotes of the next generation
is also p∗i (2.1), so that allele frequencies evolve according to the selection dynamics
p0i = pi
wi
,
w̄
i ∈ I.
(2.6)
This recurrence equation preserves the relation
X
pi = 1
i
and describes the evolution of allele frequencies at a single autosomal locus in a diploid
population. We view the selection dynamics (2.6) as a (discrete) dynamical system on
the simplex
X
>
I
SI = p = (p1 , . . . , pI ) ∈ R : pi ≥ 0 for every i ∈ I ,
pi = 1 .
(2.7)
i
Although selection destroys Hardy-Weinberg proportions, random mating re-establishes
them. Therefore, (2.6) is sufficient to study the evolutionary dynamics.
4
Throughout, the superscript
>
denotes vector or matrix transposition.
9
The right-hand side of (2.6) remains unchanged if every wij is multiplied by the same
constant. This is very useful because it allows to rescale the fitness parameters according
to convenience (also their number is reduced by one). Therefore, we will usually consider
relative fitnesses and not absolute fitnesses.
Fitnesses are said to be multiplicative if constants vi exist such that
wij = vi vj
for every i, j. Then wi = vi v̄, where v̄ =
P
i
(2.8)
vi pi , and w̄ = v̄ 2 . Therefore, (2.6) simplifies
to
vi
, i ∈ I,
(2.9)
v̄
which can be solved explicitly because it is equivalent to the linear system x0i = vi xi . It is
p0i = pi
easy to show that (2.9) also describes the dynamics of a haploid population if the fitness
vi is assigned to allele Ai .
Fitnesses are said to be additive if constants vi exist such that
wij = vi + vj
(2.10)
P
for every i, j. Then wi = vi +v̄, where v̄ = i vi pi , and w̄ = 2v̄. Although this assumption
is important (it means absence of dominance; see Sect. 2.3), it does not yield an explicit
solution of the selection dynamics.
Example 2.1. Selection is very efficient. We assume (2.8). Then the solution of (2.9) is
pi (0)vit
pi (t) = P
.
t
j pj (0)vj
(2.11)
Suppose that there are only two alleles, A1 and A2 . If A1 is the wild type and A2 is a
new beneficial mutant, we may set (without loss of generality!) v1 = 1 and v2 = 1 + s.
Then we obtain from (2.11):
p2 (t)
p2 (0)
=
p1 (t)
p1 (0)
v2
v1
t
=
p2 (0)
(1 + s)t .
p1 (0)
(2.12)
Thus, A2 increases exponentially relative to A1 .
For instance, if s = 0.5, then after 10 generations the frequency of A2 has increased by
a factor of (1 + s)t = 1.510 ≈ 57.7 relative to A2 . If s = 0.05 and t = 100, this factor is
(1 + s)t = 1.05100 ≈ 131.5. Therefore, slight fitness differences may have a big long-term
effect, in particular, since 100 generations are short on an evolutionary time scale.
10
An important property of (2.6) is that mean fitness is nondecreasing along trajectories
(solutions), i.e.,
w̄0 = w̄(p0 ) ≥ w̄(p) = w̄ ,
(2.13)
and equality holds if and only if p is an equilibrium.5
A particularly elegant proof was provided by Kingman (1961). As noted by Nagylaki
(1977), (2.13) follows immediately from an inequality of Baum and Eagon (1967) by noting
that (2.6) can be written as
p0i
∂ w̄
= pi
∂pi
X
j
pj
∂ w̄
∂pj
because ∂ w̄/∂pi = 2wi .
The statement (2.13) is closely related to Fisher’s Fundamental Theorem of Natural
Selection, which Fisher (1930) formulated as follows:
“The rate of increase in fitness of any organism at any time is equal
to its genetic variance in fitness at that time.”
For recent discussion, see Ewens (2011) and Bürger (2011).
In mathematical terms, (2.13) shows that w̄ is a Lyapunov function. This has a
number of important consequences. For instance, complex dynamical behavior such as
limit cycles or chaos can be excluded. All trajectories approach the set of points p ∈ SI
that are maxima of w̄. This is a subset of the set of equilibria. From (2.6) it is obvious
that the equilibria are precisely the solutions of
pi (wi − w̄) = 0 for every i ∈ I .
(2.14)
We call an equilibrium internal, or fully polymorphic, if pi > 0 for every i (all alleles are
present). The I equilibria defined by pi = 1 are called monomorphic because only one
allele is present.
The following result summarizes a number of important properties of the selection
dynamics. Proofs and references to the original literature may be found in Bürger (2000,
Chap. I.9); see also Lyubich (1992, Chap. 9).
Theorem 2.2. 1. If an isolated internal equilibrium exists, then it is uniquely determined.
2. p̂ is an equilibrium if and only if p̂ is a critical point of the restriction of mean
fitness w̄(p) to the minimal subsimplex of SI that contains the positive components of p̂.
p is called an equilibrium, or fixed point, of the recurrence equation p0 = f (p) if f (p) = p. We use
the term equilibrium point to emphasize that we consider an equilibrium that is a single point. The term
equilibrium may also refer to a (connected) manifold of equilibrium points.
5
11
3. If the number of equilibria is finite, then it is bounded above by 2I − 1.
4. An internal equilibrium is asymptotically stable if and only if it is an isolated local
maximum of w̄. Moreover, it is isolated if and only if it is hyperbolic (i.e., the Jacobian
has no eigenvalues of modulus 1).
5. An equilibrium point is stable if and only if it is a local, not necessarily isolated,
maximum of w̄.
6. If an asymptotically stable internal equilibrium exists, then every orbit starting in
the interior of SI converges to that equilibrium.
7. If an internal equilibrium exists, it is stable if and only if, counting multiplicities,
the fitness matrix W = (wij ) has exactly one positive eigenvalue.
8. If the matrix W has i positive eigenvalues, at least (i − 1) alleles will be absent at a
stable equilibrium.
9. Every orbit converges to one of the equilibrium points (even if stable manifolds of
equilibria exist).
2.3
Two alleles and the role of dominance
For the purpose of illustration, we work out the special case of two alleles. We write p
and 1 − p instead of p1 and p2 . Further, we use relative fitnesses and assume
w11 = 1 , w12 = 1 − hs , w22 = 1 − s ,
(2.15)
where s is called the selection coefficient and h describes the degree of dominance. We
assume s > 0.
The allele A1 is called dominant if h = 0, partially dominant if 0 < h < 21 , recessive if
h = 1, and partially recessive if
1
2
< h < 1. No dominance refers to h = 12 . Absence of
dominance is equivalent to additive fitnesses (2.10). If h < 0, there is overdominance or
heterozygote advantage. If h > 1, there is underdominance or heterozygote inferiority.
From (2.4), the marginal fitnesses of the two alleles are
w1 = 1 − hs + hsp and w2 = 1 − s + s(1 − h)p
and, from (2.3), the mean fitness is
w̄ = 1 − s + 2s(1 − h)p − s(1 − 2h)p2 .
It is easily verified that the allele-frequency change from one generation to the next can
12
Schematic selection dynamics with two alleles
0 < s < 1, h > 1
0 < s < 1, h < 0
0 < s < 1, 0 ≤ h ≤ 1
0 < s < 1, 0 ≤ h ≤ 1
p
0
1
Figure 2.1: Convergence patterns for selection with two alleles.
be written as
p(1 − p) dw̄
2w̄
dp
p(1 − p)s
=
[1 − h − (1 − 2h)p] .
w̄
∆p = p0 − p =
(2.16a)
(2.16b)
There exists an internal equilibrium if and only if h < 0 (overdominance) or h > 1
(underdominance). It is given by
p̂ =
1−h
.
1 − 2h
(2.17)
If dominance is intermediate, i.e., if 0 ≤ h ≤ 1, then (2.16) shows that ∆p > 0 if
0 < p < 1, hence p = 1 is globally asymptotically stable.
If h < 0 or h > 1, we write (2.16) in the form
∆p =
sp(1 − p)
(1 − 2h)(p̂ − p) .
w̄
(2.18)
In the case of overdominance (h < 0), we have 0 < sp(1 − p)(1 − 2h)/w̄ < 1 if 0 < p < 1,
hence p̂ is globally asymptotically stable and convergence is monotonic. If h > 1, then the
monomorphic equilibria p = 0 and p = 1 each are asymptotically stable and p̂ is unstable.
The three possible convergence patterns are shown in Figure 2.1. Figure 2.2 demonstrates that the degree of (intermediate) dominance strongly affects the rate of spread of
an advantageous allele.
13
Figure 2.2: Selection of a dominant (h = 0, solid line), intermediate (h = 1/2, dashed), and
recessive (h = 1, dash-dotted) allele. The initial frequency is p0 = 0.005 and the selective
advantage is s = 0.05. If the advantageous allele is recessive, its initial rate of increase is
vanishingly small because the frequency p2 of homozygotes is extremely low when p is small.
However, only homozygotes are ‘visible’ to selection.
2.4
The continuous-time selection model
Most higher animal species have overlapping generations because birth and death occur
continuously in time. This, however, may lead to substantial complications if one wishes
to derive a continuous-time model from biological principles. By contrast, discrete-time
models can frequently be derived straightforwardly from simple biological assumptions.
If evolutionary forces are weak, a continuous-time version can usually be obtained as an
approximation to the discrete-time model.
A rigorous derivation of the differential equations describing gene-frequency change
under selection in a diploid population with overlapping generations is a formidable task
and requires a complex model involving age structure (see Nagylaki 1992, Chap. 4.10).
Here, we simply state the system of differential equations and justify it in an alternative
way.
In a continuous-time model, the (Malthusian) fitness mij of a genotype Ai Aj is defined
as its birth rate minus its death rate. Then the marginal fitness of allele Ai is
mi =
X
j
14
mij pj ,
the mean fitness of the population is
X
X
m̄ =
mi pi =
mij pi pj ,
i
i,j
and the dynamics of allele frequencies becomes
ṗi =
dpi
= pi (mi − m̄) ,
dt
i ∈ I .6
(2.19)
This is the analogue of the discrete-time selection dynamics (2.6). Its state space is again
the simplex SI . The equilibria are obtained from the condition ṗi = 0 for every i. We
note that (2.19) is a so-called replicator equation (see Hofbauer and Sigmund 1998).
If we set
wij = 1 + smij
for every i, j ∈ I ,
(2.20)
where s > 0 is (sufficiently) small, the difference equation (2.6) and the differential equation (2.19) have the same equilibria. This is obvious upon noting that (2.20) implies
wi = 1 + smi and w̄ = 1 + sm̄.
Following Nagylaki (1992, p. 99), we approximate the discrete model (2.6) by the
continuous model (2.19) under the assumption of weak selection, i.e., small s in (2.20).
We rescale time according to t = bτ /sc, where b c denotes the closest smaller integer.
Then s may be interpreted as generation length and, for pi (t) satisfying the difference
equation (2.6), we write πi (τ ) = pi (t). Then we obtain formally
dπi
1
1
= lim [πi (τ + s) − πi (τ )] = lim [pi (t + 1) − pi (t)] .
s↓0 s
s↓0 s
dτ
From (2.6) and (2.20), we obtain pi (t + 1) − pi (t) = spi (t)(mi − m̄)/(1 + sm̄). Therefore,
π̇i = πi (mi − m̄) and ∆pi ≈ sπ̇i = spi (mi − m̄). We note that (2.6) is essentially the Euler
scheme for (2.19).
The exact continuous-time model reduces to (2.19) only if the mathematically inconsistent assumption is imposed that Hardy-Weinberg proportions apply for every t which
is generally not true. Under weak selection, however, deviations from Hardy-Weinberg
decay to order O(s) after a short period of time (Nagylaki 1992).
One of the advantages of models in continuous time is that they lead to differential
equations, and usually these are easier to analyze because the formalism of calculus is
available. An example for this is that, in continuous time, (2.13) simplifies to
˙ ≥ 0,
m̄
6
Throughout, we use a dot, ˙ , to indicate derivatives with respect to time.
15
(2.21)
which is much easier to prove than (2.13):
X
X
X
X
˙ =2
m̄
mij pj ṗi = 2
mi ṗi = 2
(m2i − m̄2 )pi = 2
(mi − m̄)2 pi .
i,j
i
i
i
Remark 2.3. The allele-frequency dynamics (2.19) can be written as a (generalized)
gradient system (Svirezhev 1972, Shahshahani 1979):
>
∂ m̄
∂ m̄
ṗ = Gp grad m̄ = Gp
,...,
.
∂p1
∂pn
(2.22)
Here, Gp = (g ij ) is a quadratic (covariance) matrix, where
g ij = Cov(fi , fj ) = 12 pi (δij − pj )
and
fi (Ak Al ) =


1
1
2

0
if k = l = i ,
if k 6= l and k = i or l = i ,
otherwise .
(2.23)
(2.24)
An equivalent formulation is the following covariance form:
ṗi = Cov(fi , m),
(2.25)
where m is interpreted as the random variable m(Ak Al ) = mkl (Li 1967). It holds under
much more general circumstances (Price 1970, Lessard 1997).
3
The general migration-selection model
We assume a population of diploid organisms with discrete, nonoverlapping generations.
This population is subdivided into Γ demes (niches). Viability selection acts within each
deme and is followed by adult migration (dispersal). After migration random mating
occurs within each deme. We assume that the genotype frequencies are the same in
both sexes (e.g., because the population is monoecious). We also assume that, in every
deme, the population is so large that gene and genotype frequencies may be treated as
deterministic, i.e., we ignore random genetic drift.
3.1
The recurrence equations
As before, we consider a single locus with I alleles Ai (i ∈ I). Throughout, we use letters
i, j to denote alleles, and greek letters α, β to denote demes. We write G = {1, . . . , Γ} for
the set of all demes. The presentation below is based on Chapter 6.2 of Nagylaki (1992).
16
We denote the frequency of allele Ai in deme α by pi,α . Therefore, we have
X
pi,α = 1
(3.1)
i
for every α ∈ G. Because selection may vary among demes, the fitness (viability) wij,α
of an Ai Aj individual in deme α may depend on α. The marginal fitness of allele Ai in
deme α and the mean fitness of the population in deme α are
wi,α =
X
wij,α pj,α
and w̄α =
j
X
wij,α pi,α pj,α ,
(3.2)
i,j
respectively.
Next, we describe migration. Let m̃αβ denote the probability that an individual in deme
α migrates to deme β, and let mαβ denote the probability that an (adult) individual in
deme α immigrated from deme β. The Γ × Γ matrices
M̃ = (m̃αβ ) and M = (mαβ )
(3.3)
are called the forward and backward migration matrices, respectively. Both matrices are
stochastic, i.e., they are nonnegative and satisfy
X
β
m̃αβ = 1 and
X
mαβ = 1 for every α .
(3.4)
β
Given the backward migration matrix and the fact that random mating within each
demes does not change the allele frequencies, the allele frequencies in the next generation
are
X
p0i,α =
mαβ p∗i,β ,
(3.5a)
β
where
p∗i,α = pi,α
wi,α
w̄α
(3.5b)
describes the change due to selection alone; cf. (2.6). These recurrence equations define a
dynamical system on the Γ-fold Cartesian product SΓI of the simplex SI . The investigation
of this dynamical system, along with biological motivation and interpretation of results,
is one of the main purposes of this survey.
The difference equations (3.5) require that the backward migration rates are known. In
the following, we derive their relation to the forward migration rates and discuss conditions
when selection or migration do not change the deme proportions.
17
3.2
The relation between forward and backward migration rates
To derive this relation, we describe the life cycle explicitly. It starts with zygotes on
which selection acts (possibly including population regulation). After selection adults
migrate and usually there is population regulation after migration (for instance because
the number of nesting places is limited). By assumption, population regulation does not
change genotype frequencies. Finally, there is random mating and reproduction, which
neither changes gene frequencies (Section 2.1) nor deme proportions. The respective
proportions of zygotes, pre-migration adults, post-migration adults, and post-regulation
0
adults in deme α are cα , c∗α , c∗∗
α , and cα :
Zygote
-
Adult
selection
cα , pi,α
Adult
migration
-
Adult
regulation
0
c∗∗
α , pi,α
c∗α , p∗i,α
-
Zygote
reproduction
c0α , p0i,α
c0α , p0i,α
Because no individuals are lost during migration, the following must hold:
c∗∗
β =
X
c∗α
X
c∗α m̃αβ ,
(3.6a)
c∗∗
β mβα .
(3.6b)
α
=
β
The (joint) probability that an adult is in deme α and migrates to deme β can be expressed
in terms of the forward and backward migration rates as follows:
c∗α m̃αβ = c∗∗
β mβα .
(3.7)
Inserting (3.6a) into (3.7), we obtain the desired connection between the forward and the
backward migration rates:
mβα
c∗α m̃αβ
P
=
.
∗
γ cγ m̃γβ
(3.8)
Therefore, if M̃ is given, an ansatz for the vector c∗ = (c∗1 , . . . , c∗Γ )> in terms of c =
(c1 , . . . , cΓ )> is needed to compute M (as well as a hypothesis for the variation, if any, of
c).
Two frequently used assumptions are the following.
1) Soft selection. This assumes that the fraction of adults in every deme is fixed, i.e.,
c∗α = cα
for every α ∈ G .
18
(3.9)
This may be a good approximation if the population is regulated within each deme, e.g.,
because individuals compete for resources locally (Dempster 1955).
2) Hard selection. Following Dempster (1955), the fraction of adults will be proportional to mean fitness in the deme if the total population size is regulated. This has been
called hard selection and is defined by
c∗α = cα w̄α /w̄ ,
(3.10)
X
(3.11)
where
w̄ =
cα w̄α
α
is the mean fitness of the total population.
Essentially, these two assumptions are at the extremes of a broad spectrum of possibilities. Soft selection will apply to plants; for animals many schemes are possible.
Under soft selection, (3.8) becomes
cα m̃αβ
mβα = P
.
γ cγ m̃γβ
(3.12)
As a consequence, if c is constant (c0 = c), M is constant if and only if M̃ is constant.
If there is no population regulation after migration, then c will generally depend on time
because (3.6a) yields c0 = c∗∗ = M̃ > c. Therefore, the assumption of constant deme
proportions, c0 = c, will usually require that population control occurs after migration.
∗
A migration pattern that does not change deme proportions (c∗∗
α = cα ) is called conservative. Under this assumption, (3.7) yields
c∗α m̃αβ = c∗β mβα
(3.13)
and, by stochasticity of M and M̃ , we obtain
X
X
c∗β mβα .
c∗β =
c∗α m̃αβ and c∗α =
α
(3.14)
β
If there is soft selection and the deme sizes are equal (c∗α = cα ≡ constant), then mαβ =
m̃βα .
Remark 3.1. Conservative migration has two interesting special cases.
1) Dispersal is called reciprocal if the number of individuals that migrate from deme α
to deme β equals the number that migrate from β to α:
c∗α m̃αβ = c∗β m̃βα .
19
(3.15)
∗
If this holds for all pairs of demes, then (3.6a) and (3.4) immediately yield c∗∗
β = cβ .
From (3.7), we infer mαβ = m̃αβ , i.e., the forward and backward migration matrices are
identical.
2) A migration scheme is called doubly stochastic if
X
m̃αβ = 1 for every α .
(3.16)
α
∗
If demes are of equal size, then (3.6a) shows that c∗∗
α = cα . Hence, with equal deme sizes
a doubly stochastic migration pattern is conservative. Under soft selection, deme sizes
remain constant without further population regulation. Hence, mαβ = m̃βα and M is also
doubly stochastic.
Doubly stochastic migration patterns arise naturally if there is a periodicity, e.g., because the demes are arranged in a circular way. If we posit equal deme sizes and homogeneous migration, i.e., m̃αβ = m̃β−α so that migration rates depend only on distance, then
the backward migration pattern is also homogeneous because mαβ = m̃βα = m̃α−β and,
hence, depends only on β − α. If migration is symmetric, m̃αβ = m̃βα , and the deme sizes
are equal, then dispersion is both reciprocal and doubly stochastic.
3.3
Important special migration patterns
We introduce three migration patterns that play an important role in the population
genetics and ecological literature.
Example 3.2. Random outbreeding and site homing, or the Deakin (1966) model. This
model assumes that a proportion µ ∈ [0, 1] of individuals in each deme leaves their deme
and is dispersed randomly across all demes. Thus, they perform outbreeding whereas
a proportion 1 − µ remains at their home site. If c∗∗
α is the proportion (of the total
population) of post-migration adults in deme α, then the forward migration rates are
∗∗
defined by m̃αβ = µc∗∗
β if α 6= β, and m̃αα = 1 − µ + µcα . If µ = 0, migration is absent;
if µ = 1, the Levene model is obtained (see below). Because this migration pattern is
reciprocal, M = M̃ holds.
To prove that migration in the Deakin model satisfies (3.15), we employ (3.7) and find
mβα =
c∗α
m̃αβ
c∗∗
β


µc∗α
= c∗β
∗

 c∗∗ (1 − µ) + µcβ
β
20
if α 6= β
if α = β .
(3.17)
From this we deduce
1=
X
mβα =
α
X
µc∗α +
α6=β
c∗β
c∗β
∗
=
µ
·
1
+
(1
−
µ)
(1
−
µ)
+
µc
,
β
c∗∗
c∗∗
β
β
(3.18)
∗
which immediately yields c∗∗
β = cβ for every β provided µ < 1. Therefore, we obtain
m̃βα = µc∗α if α 6= β and c∗β m̃βα = c∗β µc∗α = c∗α m̃αβ , i.e., reciprocity.
We will always assume soft selection in the Deakin model, i.e., c∗α = cα . Thus, for a
given (probability) vector c = (c1 , . . . , cΓ )> , the single parameter µ is sufficient to describe
the migration pattern:
mβα = m̃βα
(
µcα
=
1 − µ + µcβ
if α 6= β
if α = β .
(3.19)
Example 3.3. The Levene (1953) model assumes soft selection and
mαβ = cβ .
(3.20)
Thus, dispersing individuals are distributed randomly across all demes in proportion to
the deme sizes. In particular, migration is independent of the deme of origin and M = M̃ .
Alternatively, the Levene model could be defined by m̃αβ = µβ , where µβ > 0 are
P
constants satisfying β µβ = 1. Then (3.8) yields mαβ = c∗β for every α, β ∈ G. With
soft selection, we get mαβ = cβ . This is all we need if demes are regulated to constant
proportions. But the proportions remain constant even without regulation, for (3.6a)
gives c0α = c∗∗
α = µα . This yields the usual interpretation µα = cα (Nagylaki 1992, Sect.
6.3).
Example 3.4. In the linear stepping-stone model the demes are arranged in a linear
order and individuals can reach only one of the neighboring demes. It is an extreme case
among migration patterns exhibiting isolation by distance, i.e., patterns in which migration diminishes with the distance from the parental deme. In the classical homogeneous
version, the forward migration matrix is


1−m
m
0
...
0
 m
1 − 2m m
0 


 ..

.
.
.
.
M̃ =  .
.
.
.


 0
m 1 − 2m
m 
0
...
0
m
1−m
21
(3.21)
We leave it to the reader to derive the backward migration matrix using (3.8). It is a
special case of the following general tridiagonal form:


n1 r1
0
...
0
 q2 n2 r2
0 


 ..

.
.
.
.
M =.
,
.
.


0
qΓ−1 nΓ−1 rΓ−1 
0 ...
0
qΓ
nΓ
(3.22)
where nα ≥ 0 and qα + nα + rα = 1 for every α, qα > 0 for α ≥ 2, rα > 0 for α ≤ Γ − 1, and
q1 = rΓ = 0. This matrix admits variable migration rates between neighboring demes.
If all deme sizes are equal, the homogeneous matrix (3.21) satisfies M = M̃ , and each
deme exchanges a fraction m of the population with each of its neighboring demes. The
stepping-stone model has been used as a starting point to derive the partial differential
equations for selection and dispersal in continuous space (Nagylaki 1989). Also circular
and infinite variants have been investigated.
Juvenile migration is of importance for many marine organisms and plants, where seeds
disperse. It can be treated in a similar way as adult migration. Also models with both
juvenile and adult migration have been studied. Some authors investigated migration
and selection in dioecious populations, as well as selection on X-linked loci (e.g., Nagylaki
1992, pp. 143, 144).
Unless stated otherwise, throughout this survey we assume that the backward migration matrix M is constant, as is the case for soft selection if deme proportions and the
forward migration matrix are constant. Then the recurrence equations (3.5) provide a
self-contained description of the migration-selection dynamics. Hence, they are sufficient
to study evolution for an arbitrary number of generations.
4
Two alleles
Of central interest is the identification of conditions that guarantee the maintenance of
genetic diversity. Often it is impossible to determine the equilibrium structure in detail
because establishing existence and, even more so, stability or location of polymorphic
equilibria is unfeasible. Below we introduce an important concept that is particularly
useful to establish maintenance of genetic variation at diallelic loci. Throughout this
section we consider a single locus with two alleles. The number of demes, Γ, can be
arbitrary.
22
4.1
Protected polymorphism
There is a protected polymorphism (Prout 1968) if, independently of the initial conditions
are, a polymorphic population cannot become monomorphic. Essentially, this requires
that if an allele becomes very rare, its frequency must increase. In general, a protected
polymorphism is neither necessary nor sufficient for the existence of a stable polymorphic
equilibrium. For instance, on the one hand, if there is an internal limit cycle that attracts
all solutions, then there is a protected polymorphism. On the other hand, if there are
two internal equilibria, one asymptotically stable, the other unstable, then selection may
remove one of the alleles if sufficiently rare. A generalization of this concept to multiple
alleles would correspond to the concept of permanence often used in ecological models
(e.g., Hofbauer and Sigmund 1998).
Because we consider only two alleles, we can simplify the notation. We write pα = p1,α
for the frequency of allele A1 in deme α (and 1 − pα for that of A2 in deme α). Let
p = (p1 , . . . , pΓ )> denote the vector of allele frequencies. Instead of using the fitness
assignments w11,α , w12,α , and w22,α , it will be convenient to scale the fitness of the three
genotypes in deme α as follows
A1 A1
xα
A1 A2
1
A2 A2
yα
(4.1)
(xα , yα ≥ 0). This can be achieved by setting xα = w11,α /w12,α and yα = w22,α /w12,α ,
provided w12,α > 0.
With these fitness assignments, one obtains
w1,α = 1 − pα + xα pα
and w̄α = xα p2α + 2pα (1 − pα ) + yα (1 − pα )2 ,
(4.2)
and the migration-selection dynamics (3.5) becomes
p∗α = pα w1,α /w̄α
X
p0α =
mαβ p∗β .
(4.3a)
(4.3b)
β
We consider this as a (discrete) dynamical system on [0, 1]Γ .
We call allele A1 protected if it cannot be lost. Thus, it has to increase in frequency
if rare. In mathematical terms this means that the monomorphic equilibrium p = 0 must
be unstable. To derive a sufficient condition for instability of p = 0, we linearize (4.3)
at p = 0. If yα > 0 for every α (which means that A2 A2 is nowhere lethal), a simple
23
calculation shows that the Jacobian of (4.3a),
∗ ∂pα D=
∂pβ ,
(4.4)
p=0
is a diagonal matrix with (nonzero) entries dαα = yα−1 . Because (4.3b) is linear, the
linearization of (4.3) is
p0 = Qp ,
where Q = M D ,
(4.5)
i.e., qαβ = mαβ /yβ .
To obtain a simple criterion for protection, we assume that the descendants of individuals in every deme be able eventually to reach every other deme. Mathematically,
the appropriate assumption is that M is irreducible. Then Q is also irreducible and it is
nonnegative. Therefore, the Theorem of Perron and Frobenius (e.g., Seneta 1981) implies
the existence of a uniquely determined eigenvalue λ0 > 0 of Q such that |λ| ≤ λ0 holds
for all eigenvalues of Q. In addition, there exists a strictly positive eigenvector pertaining
to λ0 which, up to multiplicity, is uniquely determined. As a consequence,
A1 is protected if λ0 > 1 and A1 is not protected if λ0 < 1
(4.6)
(if λ0 = 1, then stability cannot be decided upon linearization). This maximal eigenvalue
satisfies
min
X
α
qαβ ≤ λ0 ≤ max
X
α
β
qαβ ,
(4.7)
β
with equality if and only if all the row sums are the same.
Example 4.1. Suppose that A2 A2 is at least as fit as A1 A2 in every deme and more fit in
at least one deme, i.e., yα ≥ 1 for every α and yβ > 1 for some β. Then qαβ = mαβ /yβ ≤
mαβ for every β. Because M is irreducible, there is no β such that mαβ = 0 for every α.
P
P
Therefore, the row sums β qαβ = β mαβ /yβ in (4.7) are not all equal to one, and we
obtain
λ0 < max
α
X
qαβ ≤ max
α
β
X
mαβ = 1 .
(4.8)
β
Thus, A1 is not protected, and this holds independently of the choice of the xα , or w11,α .
It can be shown similarly that A1 is protected if A1 A2 is favored over A2 A2 in at least
one deme and is nowhere less fit than A2 A2 .
One obtains the condition for protection of A2 if, in (4.6), A1 is replaced by A2 and λ0
is the maximal eigenvalue of the matrix with entries mαβ /xβ . Clearly, there is a protected
polymorphism if both alleles are protected.
24
In the case of complete dominance the eigenvalue condition (4.6) cannot be satisfied.
Consider, for instance, protection of A1 if A2 is dominant, i.e., yα = 1 for every α. Then
P
P
qαβ = mαβ , β qαβ = β mαβ = 1, and λ0 = 1. This case is treated in Section 6.2 of
Nagylaki (1992).
4.2
Two demes
It will be convenient to set
xα = 1 − rα
and yα = 1 − sα ,
(4.9)
where rα ≤ 1 and sα ≤ 1 for every α ∈ {1, 2}. We write the backward migration matrix
as
1 − m1
m1
M=
,
(4.10)
m2
1 − m2
where 0 < mα < 1 for every α ∈ {1, 2}.
Now we derive the condition for protection of A1 . The characteristic polynomial of Q
is given by
ϕ(x) = (1 − s1 )(1 − s2 )x2 − (2 − m1 − m2 − s1 − s2 + s1 m2 + s2 m1 )x + 1 − m1 − m2 . (4.11)
It is convex and satisfies
ϕ(0) = 1 − m1 − m2 > 0 ,
ϕ(1) = s1 s2 (1 − κ) ,
ϕ0 (1) = (1 − s1 )(m2 − s2 ) + (1 − s1 )(m1 − s1 ) ,
where
κ=
m1 m2
+
.
s1
s2
(4.12a)
(4.12b)
(4.13)
By Example 4.1, A1 is not protected if A1 A2 is less fit than A2 A2 in both demes (more
generally, if s1 ≤ 0, s2 ≤ 0, and s1 + s2 < 0). Of course, A1 will be protected if A1 A2
is fitter than A2 A2 in both demes (more generally, if s1 ≥ 0, s2 ≥ 0, and s1 + s2 > 0).
Hence, we restrict attention to the most interesting case when A1 A2 is fitter than A2 A2
in one deme and less fit in the other, i.e., s1 s2 < 0.
The Perron-Frobenius Theorem informs us that ϕ(x) has two real roots. We have to
determine when the larger (λ0 ) satisfies λ0 > 1. From (4.11) and (4.12), we infer that
this is the case if and only if ϕ(1) < 0 or ϕ(1) > 0 and ϕ0 (1) < 0. It is straightforward to
25
s2
1
m2
ț =1
m1
s1
1
0
1
Figure 4.1: The region of protection of AFig.
From Nagylaki and Lou (2008).
1 (hatched).
show that ϕ(1) > 0 and ϕ0 (1) < 0 is never ssatisfied
if s1 s2 < 0. Therefore, we conclude
2
that, if s1 s2 < 0, allele A1 is protected
ț = -1 if
1
κ < 1;
ȍ+
(4.14)
ȍ1
ȍ1
cf. Bulmer (1972). It is not
4.1 displays the region of protection
ț =1 protected if κ > 1. Figure
of A1 for given m1 and m2 .
ȍ
0
If there is no dominance -1(rα = −sα and
0 < |sα | < 11 for αs1 = 1, 2), then further
0
ȍ
simplification can be achieved. From the preceding1 paragraph the results depicted in
0 a protected polymorphism is
Figure 4.2 are obtained. The regionȍof
ȍ0
ȍ+
ț = -1
Ω+ = {(s1 , s2 ) : s1 s2 < 0 and |κ| < 1} .
-1
(4.15)
ț =1
In a panmictic population, a stable polymorphism
can not occur in the absence of
overdominance. Protection of both alleles in a subdivided population requires that selecFig. and
2
tion in the two demes is in opposite direction
sufficiently strong relative to migration.
Therefore, the study of the maintenance of polymorphism is of most interest if selection
acts in opposite direction and dominance is intermediate, i.e.,
rα sα < 0 for α = 1, 2 and s1 s2 < 0.
26
(4.16)
Fig. 1
s2
ț = -1
1
ȍ+
ȍ1
ȍ1
ț =1
ȍ0
-1
0
ȍ1
s1
1
ȍ0
ȍ0
ț = -1
ȍ+
-1
ț =1
Figure 4.2: The regions of protection of A2 only
Fig. 2 (Ω0 ), A1 only (Ω1 ), and both A1 and A2 (Ω+ )
in the absence of dominance. From Nagylaki and Lou (2008).
Example 4.2. It is illuminating to study how the parameter region in which a protected
polymorphism exists depends on the degree of dominance in the two demes. Figures
4.3 and 4.4 display the regions of protected polymorphism for two qualitatively different
scenarios of dominance. In the first, the fitnesses are given by
A1 A1
1+s
1 − as
A1 A2
A2 A2
1 + hs 1 − s
1 − has 1 + as
(4.17)
,
thus, there is deme independent degree of dominance (DIDID). In the second scenario,
the fitnesses are given by
A1 A1
1+s
1 − as
A1 A2
A2 A2
1 + hs 1 − s
1 + has 1 + as
(4.18)
.
In both cases, we assume 0 < s < 1, 0 < a < 1 (a is a measure of the selection intensity
in deme 2 relative to deme 1), and −1 ≤ h ≤ 1 (intermediate dominance).
If, in (4.17), selection is sufficiently symmetric, i.e., a > 1/(1 + 2s), there exists a
protected polymorphism for every h ≤ 1 and every m ≤ 1. Otherwise, for given a, the
critical migration rate m admitting a protected polymorphism decreases with increasing h
27
m
0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
a
Figure 4.3: Influence of the degree of dominance on the region of protected polymorphism
(shaded). The fitness scheme is (4.17) with s = 0.1, and migration is symmetric, i.e., m1 =
m2 = m. The values of h are -0.95, -0.5, 0, 0.5, 0.95 and correspond to the curves from left to
right (light shading to dark shading). A protected polymorphism is maintained in the shaded
area to the right of the respective curve. To the right of the white vertical line, a protected
polymorphism exists for every m.
m
0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
a
Figure 4.4: Influence of the degree of dominance on the region of protected polymorphism
(shaded). The fitness scheme is (4.18) with s = 0.1, and migration is symmetric, i.e., m1 =
m2 = m. The values of h are -0.75, -0.25, 0, 0.25, 0.75 and correspond to the curves from right
to left (shading from dark to light!). A protected polymorphism is maintained in the shaded
area to the right of the respective curve.
28
because this increases the average (invasion) fitness of A1 . This can be shown analytically
by studying the principal eigenvalue.
For (4.18), increasing h greatly facilitates protected polymorphism because it leads to
an increase of the average fitness of heterozygotes relative to the homozygotes. The precise
argument is as follows. If we rescale fitnesses in (4.18) according to (4.1), the matrix Q in
(4.5) has the entries qα1 = mα1 (1 + hs)/(1 − s) and qα2 = mα1 (1 + has)/(1 − as), which are
increasing in h. Therefore, the principal eigenvalues λ0 of Q increases in h (e.g., Berman
and Plemmons 1994, Chapter 1.3), and (4.6) implies that protection of A1 is facilitated
by increasing h. Because an analogous reasoning applies to A2 , the result is proved.
Indeed, the above finding on the role of dominance for the fitness scheme (4.18) is a
special case the following result of Nagylaki (personal communication). Assume fitnesses
of A1 A1 , A1 A2 , and A2 A2 in deme α are 1 + sα , 1 + hα sα , and 1, respectively, where
sα ≥ 1 and 0 ≤ hα ≤ 1. If the homozygote fitnesses are fixed, increasing the heterozygote
fitness in each deme aids the existence of a protected polymorphism. The proof follows
by essentially the same argument as above.
Example 4.3. In the Deakin model, the condition (4.14) for protection of allele A1
becomes
c2 c1
κ=µ
+
< 1,
(4.19)
s1 s2
where s1 s2 < 0. Therefore, for given s1 , s2 , and c1 , there is a critical value µ0 such that
allele A1 is protected if and only if µ < µ0 . This implies that for two diallelic demes a
protected polymorphism is favored by a smaller migration rate.
Example 4.4. In the Levene model, the condition for a protected polymorphism is
c2 c1
+
< 1 and
s1 s2
c2 c1
+
< 1.
r1 r2
(4.20)
We close this subsection with an example showing that already with two alleles and
two demes the equilibrium structure can be quite complicated.
Example 4.5. In the absence of migration, the recurrence equations for the allele frequencies p1 , p2 in the two demes are two decoupled one-locus selection dynamics of the
form (2.16). Therefore, if there is underdominance in each deme, the top convergence
pattern in Figure 2.1 applies to each deme. As a consequence, in the absence of migration, the complete two-deme system has nine equilibria, four of which are asymptotically
stable and the others are unstable. Under sufficiently weak migration all nine equilibria
29
are admissible and the four stable ones remain stable, whereas the other five are unstable.
Two of the stable equilibria are internal. For increasing migration rate, several of these
equilibria are extinguished in a sequence of bifurcations (Karlin and McGregor 1972a).
4.3
Arbitrary number of demes
The following result is a useful tool to study protection of an allele. Let I (n) and Q(n)
respectively designate the n × n unit matrix and the square matrix formed from the first
n rows and columns of Q.
Theorem 4.6 (Christiansen 1974). If there exists a permutation of demes such that
det(I (n) − Q(n) ) < 0
(4.21)
for some n, where 1 ≤ n ≤ Γ, then A1 is protected.
This theorem is sharp in the sense that if the inequality in (4.21) is reversed for every
n ≤ Γ, then A1 is not protected.
The simplest condition for protection is obtained from Theorem 4.6 by setting n = 1.
Hence, A1 is protected if an α exists such that (Deakin 1972)
mαα /yα > 1 .
(4.22)
This condition ensures that, when rare, the allele A1 increases in frequency in deme α
even if this subpopulation is the only one containing the allele. The general condition
in the above theorem ensures that A1 increases in the n subpopulations if they are the
only ones that contain it. Therefore, we get the following simple sufficient condition for
a protected polymorphism:
There exist α and β such that mαα /yα > 1 and mββ /xβ > 1.
(4.23)
If we apply these results to the Deakin model with an arbitrary number of demes,
condition (4.23) becomes
There exist α and β such that
1 − µ(1 − cα )
1 − µ(1 − cβ )
> 1 and
> 1.
yα
xβ
(4.24)
A more elaborate and less stringent condition follows from Theorem 4.6 by nice matrix
algebra:
30
Corollary 4.7 (Christiansen 1974). For the Deakin model with Γ ≥ 2 demes, the following
is a sufficient condition for protection of A1 . There exists a deme α such that
1 − yα ≥ µ
(4.25)
or, if (4.25) is violated in every deme α, then
X
cα
µ
> 1.
µ
+
y
−
1
α
α
(4.26)
If (4.25) is violated for every α and the inequality in (4.26) is reversed, then A1 is not
protected.
Corollary 4.7 can be extended to the inhomogeneous Deakin model, which allows for
different homing rates (Christiansen 1974; Karlin 1982, p. 182). Using Corollary 4.7, we
can generalize the finding from two diallelic demes that a lower degree of outbreeding is
favorable for protection of one or both alleles. More precisely, we show:
Corollary 4.8. If 0 < µ1 ≤ µ2 ≤ 1, then allele A1 is protected for µ1 if µ2 satisfies the
conditions in Corollary 4.7.
Proof. If condition (4.25) holds for µ2 , it clearly holds for µ1 . Now suppose that 1−yα < µ1
for every α (hence 1 − yα < µ2 ) and that µ2 satisfies (4.26). This is equivalent to
X cα (1 − yα ) µ1 + yα − 1 X cα (1 − yα )
=
> 0.
µ1 + y α − 1 µ2 + y α − 1
µ2 + yα − 1
α
α
(4.27)
Because (µ1 + yα − 1)/(µ2 + yα − 1) ≥ µ1 /µ2 if and only if yα > 1, we obtain
X cα (1 − yα )
µ1 X cα (1 − yα )
≥
> 0,
µ2 α µ1 + yα − 1
µ
+
y
−
1
2
α
α
(4.28)
which proves our assertion.
In general, it is not true that less migration favors the maintenance of polymorphism.
For instance, if the amount of homing (1 − µα ) varies among demes, a protected polymorphism may be destroyed by decreasing one µα (Karlin 1982, p. 128).
Example 4.9. We apply Corollary 4.7 to the Levene model. Because (4.25) can never
be satisfied if µ = 1, A1 is protected from loss if the harmonic mean of the yα is less than
one, i.e., if (Levene 1953)
y∗ =
X cα
α
yα
31
!−1
< 1.
(4.29a)
Analogously, allele A2 is protected if
X cα
xα
α
x∗ =
!−1
< 1.
(4.29b)
Jointly, (4.29a) and (4.29b) provide a sufficient condition for a protected polymorphism.
If y ∗ > 1 or x∗ > 1, then A1 or A2 , respectively, is lost if initially rare.
If A1 is recessive everywhere (yα = 1 for every α), then it is protected if (Prout 1968)
x̄ =
X
cα x α > 1 .
(4.30)
α
Therefore, a sufficient condition for a protected polymorphism is
x∗ < 1 < x̄ .
(4.31)
Whereas in the Deakin model and in its special case, the Levene model, dispersal
does not depend on the geographic distribution of niches, in the stepping-stone model it
occurs between neighboring niches. How does this affect the maintenance of a protected
polymorphism? Here is the answer:
Example 4.10. For the general stepping-stone model (3.22) with fitnesses given by (4.1),
Karlin (1982) provided the following explicit criterion for protection of A1 :
X πα X
πα > 1 ,
(4.32)
y
α
α
α
where π1 = 1 and
πα =
rα−1 rα−2 · · · r1
qα qα−1 · · · q2
(4.33)
if α ≥ 2. This result is a simple consequence of Lemma 4.13.
For the homogeneous stepping-stone model with equal demes sizes, i.e., M given by
(3.21), (4.32) simplifies to
1X 1
> 1,
Γ α yα
(4.34)
which is the same as condition (4.29) in the Levene model with equal deme sizes.
Thus, we obtain the surprising result that the conditions for protection are the same
in the Levene model and in the homogeneous stepping stone model provided all demes
have equal size.
32
4.4
The continent-island model
We consider a population living on an island. At first, we admit an arbitrary number of
alleles. Each generation, a proportion a of adults is removed by mortality or emigration
and a fraction b of migrants with constant allele frequencies q̂i is added. A simple interpretation is that of one-way migration from a continent to an island. The continental
population is in equilibrium with allele frequencies q̂i > 0. Sometimes, q̂i is interpreted as
the average frequency over (infinitely) many islands and the model is simply called island
model.
Assuming random mating on the island, the dynamics of allele frequencies pi on the
island becomes
wi
+ mq̂i ,
(4.35)
w̄
where wi is the fitness of allele Ai on the island and m = b/(1 − a + b). Thus, m is the
p0i = (1 − m)pi
fraction of zygotes with immigrant parents. Throughout we assume 0 < m < 1.
If we define uij = mq̂j and consider uij as the mutation rate from Ai to Aj , a special
case of the (multiallelic) mutation-selection model is obtained (the so-called house-ofcards model). Therefore (Bürger 2000, pp. 102-103), (4.35) has the Lyapunov function
V (p) = w̄(p)1−m
Y
pi2mq̂i .
(4.36)
i
It follows that all trajectories are attracted by the set of equilibria.
In the sequel we determine the conditions under which an ‘island allele’ persists in
the population despite immigration of other alleles from the continent. Clearly, no allele
carried to the island recurrently by immigrants can be lost.
We investigate the diallelic case and consider alleles A1 and A2 with frequencies p and
1 − p on the island, and q̂2 = 1 on the continent. Thus, all immigrants are A2 A2 . For the
fitnesses on the island, we assume
w11 = 1 + s , w12 = 1 + hs , w22 = 1 − s ,
(4.37)
where 0 < s ≤ 1 and −1 ≤ h ≤ 1. Therefore, A1 evolves according to
p0 = f (p) = (1 − m)p
w1
.
w̄
(4.38)
We outline the analysis of (4.38). Since f (1) = 1 − m < 1, this confirms that A2
cannot be lost (p = 1 is not an equilibrium). Because
f (p) = p(1 − m)
33
w12
+ O(p2 )
w22
(4.39)
as p → 0, the allele A1 is protected if
m<1−
w22
.
w12
(4.40)
Obviously, p = 0 is an equilibrium. If there is no other equilibrium, then it must be
globally asymptotically stable (as is also obvious from f (1) < 1, which implies p0 < p).
The equilibria with p 6= 0 satisfy
w̄ = (1 − m)w1 ,
(4.41)
which is quadratic in p. With the fitnesses (4.37), the solutions are
p̂± =
i
p
1 h
1 + 3h + m(1 − h) ± (1 − h)2 (1 + m)2 + 8hm(1 + 1/s) ,
4h
(4.42)
These solutions give rise to feasible equilibria if 0 ≤ p± ≤ 1. As h → 0, (4.42) gives the
correct limit,
p̂− =
1 − m/s
1+m
if h = 0.
(4.43)
We define
1+m
,
3−m
µ1 = 1 + h(1 − m) ,
(1 − h)2 (1 + m)2
µ2 = −m −
.
8h
h0 = −
(4.44a)
(4.44b)
(4.44c)
Then µ2 > µ1 if and only if h < h0 . If µ2 < µ1 , then p̂+ is not admissible. As m increases
from 0 to 1, h0 decreases from − 31 to −1. Hence, if there is no dominance or the fitter
allele A1 is (partially) dominant (0 ≤ h ≤ 1), then h > h0 . If A1 is recessive, then h < h0 .
For fixed h and s it is straightforward, but tedious, to study the dependence of p̂+ and
p̂− on m. The results of this analysis can be summarized as follows (for an illustration,
see Figure 4.5):
Theorem 4.11. (a) Let m/s < µ1 . Then 0 and p− are the only equilibria and p(t) → p̂−
as t → ∞. Thus, for sufficiently weak migration, a unique polymorphism is established
independently of the degree of dominance and the initial condition.
(b) Let m/s > µ2 and −1 ≤ h < h0 , or m/s ≥ µ1 and h0 ≤ h ≤ 1, i.e., migration
is strong relative to selection. Then there exists no internal equilibrium and p(t) → 0 as
t → 0, i.e., the continental allele A2 becomes fixed.
34
b)
1
Allele frequency, p
Allele frequency, p
a)
0
0
1
0
1
Scaled migration rate, m/s
2
0 1
Scaled migration rate, m/s
Figure 4.5: Bifurcation patterns for the one-locus continent-island model. Bold lines indicate
an asymptotically stable equilibrium, dashed lines an unstable equilibrium. Figure a displays
the case h0 ≤ h ≤ 1, and Figure b the case −1 ≤ h < h0 . The parameters are s = 0.1, and
h = 0.5 and h = −0.9 in cases a and b, respectively. In a, we have µ1 ≈ 1.41; in b, µ1 ≈ 0.11,
µ2 ≈ 0.5
(c) Let µ1 < m/s ≤ µ2 and −1 ≤ h < h0 , i.e., migration is moderately strong relative
to selection and the fitter allele is (almost) recessive. Then there are the three equilibria
0, p̂+ , and p̂− , where 0 < p̂+ < p̂− , and
p(t) → 0
if p(0) < p̂+ ,
(4.45a)
p(t) → p̂−
if p(0) > p̂+ .
(4.45b)
For a detailed proof, see Nagylaki (1992, Chapter 6.1). The global dynamics of the
diallelic continent-island model admitting evolution on the continent is derived in Nagylaki
(2009a).
Sometimes immigration from two continents is considered (Christiansen 1999). Some
authors call our general migration model with n demes the n-island model (e.g., Christiansen 1999). The following (symmetric) island model is a standard model in investigations of the consequences of random drift and mutation in finite populations:
mαα = 1 − m ,
m
mαβ =
if α 6= β .
Γ−1
(4.46a)
(4.46b)
Clearly, migration is doubly stochastic in this model and there is no isolation by distance.
In the special case when all deme sizes are equal, i.e., cα = 1/Γ, (4.46) reduces to a special
case of the Deakin model, (3.19), with m = µ(1 − Γ−1 ). In this, and only this case, the
island model is conservative and the stationary distribution of M is c = e/Γ.
35
4.5
Submultiplicative fitnesses
Here, we admit arbitrary migration patterns but assume that fitnesses are submultiplicative, i.e., the fitnesses in (4.1) satisfy
xα y α ≤ 1
(4.47)
for every α (Karlin and Campbell 1980). We recall from Section 2.2 that with multiplicative fitnesses (xα yα = 1), the diploid selection dynamics reduces to the haploid dynamics.
Throughout, we denote left principal eigenvector of M by ξ ∈ SΓ ; cf. (7.24).
Theorem 4.12 (Karlin and Campbell 1980, Result I). If Γ ≥ 2 and fitnesses are submultiplicative in both demes, then at most one monomorphic equilibrium can be asymptotically
stable and have a geometric rate of convergence.
The proof of this theorem applies the following very useful result to the conditions for
protection of A1 or A2 (Section 4.1).
Lemma 4.13 (Friedland and Karlin 1975). If M is a stochastic n × n matrix and D is
a diagonal matrix with entries di > 0 along the diagonal, then
ρ(M D) ≥
n
Y
dξi i
(4.48)
i=1
holds, where ρ(M D) is the spectral radius of M D.
Without restrictions on the migration matrix, the sufficient condition
Y
(1/yα )ξα > 1
(4.49)
α
for protection of A1 is sharp. This can be verified for a circulant stepping stone model,
when ξα = 1/Γ.
The conclusion of Theorem 4.12 remains valid under some other conditions (see Karlin
and Campbell 1980). For instance, if migration is doubly stochastic, then ξα = 1/Γ for
Q 1/Γ
every α and ρ(M D) ≥ α dα . Therefore, the compound condition
Y
xα y α < 1
α
implies the conclusion of the theorem.
Here is another interesting result:
36
(4.50)
Theorem 4.14 (Karlin and Campbell 1980, Result III). If Γ = 2 and fitnesses are
multiplicative, then there exists a unique asymptotically stable equilibrium which is globally
attracting.
Because the proof given by Karlin and Campbell (1980) is erroneous, we present a
corrected proof (utilizing their main idea).
Proof of Theorem 4.14. Because fitnesses are multiplicative, we can treat the haploid
model. Without loss of generality we assume that the fitnesses of alleles A1 and A2
in deme 1 are s1 ≥ 1 and 1, respectively, and in deme 2 they are s2 (0 < s2 < 1) and 1.
(We exclude the case when one allele is favored in both demes, hence goes to fixation.)
Let the frequency of A1 in demes 1 and 2 be p1 and p2 , respectively. Then the recursion
becomes
s1 p1
s2 p2
+ m1
,
s1 p1 + 1 − p1
s2 p2 + 1 − p 2
s2 p2
s1 p1
p02 = m2
+ (1 − m2 )
.
s1 p1 + 1 − p1
s2 p2 + 1 − p 2
p01 = (1 − m1 )
(4.51a)
(4.51b)
Solving p01 = p1 for p2 , we obtain
p2 =
p1 [(1 − p1 )(1 − s1 ) + s1 m1 ]
.
p1 (1 − p1 )(1 − s1 )(1 − s2 ) + m1 (s2 + s1 p1 − s2 p1 )
(4.52)
Substituting this into p02 = p2 , we find that every equilibrium must satisfy
p1 (1 − p1 )(1 − p1 + s1 p1 )A(p1 ) = 0 ,
(4.53)
where
A(p1 ) = m1 [(1 − s1 )(1 − s2 + m2 s2 ) + m1 s1 (1 − s2 )]
− (1 − s1 ) {(1 − m2 )(1 − s1 )(1 − s2 ) + m1 [1 − s2 (1 − m2 ) + s1 (1 − m2 − s2 )]} p1
+ (1 − m2 )(1 − s1 )2 (1 − s2 )p21 .
(4.54a)
We want to show that A(p1 ) has always two zeros, because then zeros in (0, 1) can emerge
only by bifurcations through either p1 = 0 or p1 = 1. Therefore, we calculate the discriminant
D = (1 − m2 )2 σ 2 τ 2 + 2m1 (1 − m2 )στ [m2 (2 + σ − τ ) − στ ]
+ m21 [m22 (σ + τ )2 − 2m2 σ(2 + σ − τ )τ + σ 2 τ 2 ] ,
37
(4.55a)
where we set s1 = 1 + σ and s2 = 1 − τ with σ ≥ 0 and 0 < τ < 1.
Now we consider D as a (quadratic) function in m1 . To show that D(m1 ) > 0 if
0 < m1 < 1, we compute
D(0) = (1 − m2 )2 σ 2 τ 2 > 0 , D(1) = m22 [τ − σ(1 − τ )]2 > 0 ,
(4.56)
D0 (0) = 2(1 − m2 )στ [m2 (2 + σ − τ ) − στ ] .
(4.57)
and
We distinguish two cases.
1. If m2 (2 + σ − τ ) ≥ στ , then D0 (0) ≥ 0 and there can be no zero in (0, 1). (If D
is concave, then D(m1 ) > 0 on [0, 1] because D(1) > 0; if D is convex, then D0 (m1 ) > 0
holds for every m1 > 0, hence D(m1 ) > 0 on [0, 1].)
2. If m2 (2 + σ − τ ) < στ , whence D0 (0) < 0 and m2 <
στ
,
2+σ−τ
we obtain
D0 (1) = 2m2 [−στ (2 + σ − τ − στ ) + m2 (σ 2 (1 − τ ) + τ 2 + στ 2 )]
4σ(1 + σ)(1 − τ )τ
< −2m2
< 0.
2+σ−τ
(4.58a)
Therefore, D has no zero in (0, 1). Thus, we have shown that A(p1 ) = 0 has always two
real solutions.
For the rest of the proof we can follow Karlin and Campbell: If s1 = 1 and s2 < 1,
there are no polymorphic equilibria, p1 = 0 is stable and p1 = 1 is unstable. As s1
increases from 1, a bifurcation event at p1 = 0 occurs before p1 = 1 becomes stable.
At this bifurcation event, p1 = 0 becomes unstable and a stable polymorphic equilibrium
bifurcates off p1 = 0. Because the constant term m1 [(1 − s1 )(1 − s2 + m2 s2 ) + m1 s1 (1 − s2 )]
in A(p1 ) is linear in s1 , no further polymorphic equilibria can appear as s1 increases until
p = 1 becomes stable, which then remains stable for larger s1 .
Karlin and Campbell (1980) pointed out that the dynamics (4.51) has the following
monotonicity property: If p1 < q1 and p2 < q2 , then p01 < q10 and p02 < q20 . By considering
1 − p2 and 1 − q2 , p1 < q1 and p2 > q2 implies p01 < q10 and p02 > q20 . In particular, if p1 < p01
and p2 < p02 , then p01 < p001 and p02 < p002 . Therefore, this monotonicity property implies
global monotone convergence to the asymptotically stable equilibrium.
Remark 4.15. Because the monotonicity property used in the above proof holds for an
arbitrary number of demes and for selection on diploids, Karlin and Campbell (1980)
stated that “periodic or oscillating trajectories seem not to occur” (in diallelic models).
However, the argument in the above proof cannot be extended to more than two demes
38
and Akin (personal communication) has shown that for three diallelic demes unstable
periodic orbits can occur in the corresponding continuous-time model (cf. Section 6.2).
For two diallelic demes, however, every trajectory converges to an equilibrium.
Karlin and Campbell (1980) showed that for some migration patterns, Theorem 4.14
remains true for an arbitrary number of demes. The Levene model is one such example. In
fact, Theorem 5.16 is a much stronger result. It is an open problem whether Theorem 4.14
holds for arbitrary migration patterns if there are more than two demes. An interesting
and related result is the following.
Theorem 4.16 (Karlin and Campbell 1980, Result IV). Let Γ ≥ 2 and let M be arbitrary but fixed. If there exists a unique, globally attracting equilibrium under multiplicative
fitnesses, then there exists a unique, globally attracting equilibrium for arbitrary submultiplicative fitnesses.
5
The Levene model
This is a particularly simple model to examine the consequences of spatially varying
selection for the maintenance of genetic variability. It can also be interpreted as a model
of frequency-dependent selection (e.g., Wiehe and Slatkin 1998). From the definition
(3.20) of the Levene model and equation (3.5a), we infer
X
p0i,α =
cβ p∗i,β ,
(5.1)
β
which is independent of α. Therefore, after one round of migration allele frequencies are
the same in all demes, whence it is sufficient to study the dynamics of the vector of allele
frequencies
p = (p1 , . . . , pI )> ∈ SI .
(5.2)
The migration-selection dynamics (3.5) simplifies drastically and yields the recurrence
equation
p0i = pi
X
α
cα
wi,α
w̄α
(i ∈ I)
(5.3)
for the evolution of allele frequencies in the Levene model. Therefore, there is no population structure in the Levene model although the population experiences spatially varying
selection. In particular, distance does not play any role. The dynamics (5.3) is also obtained if, instead of the life cycle in Section 3.2, it is assumed that adults form a common
mating pool and zygotes are distributed randomly to the demes.
39
Remark 5.1. 1. In the limit of weak selection, the Levene model becomes equivalent to
panmixia. Indeed, with wij,α = 1 + srij,α , an argument analogous to that below (2.20)
shows that the resulting dynamics is of the form
X
ṗi = pi
cα (wi,α − w̄α ) = pi (zi − z̄) ,
(5.4)
α
P
P P
P
where zi =
j zij pj =
j
α cα rij,α pj is linear in p and z̄ =
i zi pi . Here, zij =
P
α cα rij,α is the spatially averaged selection coefficient of Ai Aj (Nagylaki and Lou 2001).
Therefore, the Levene model is of genuine interest only if selection is strong. For arbitrary migration, the weak-selection limit generally does not lead to a panmictic dynamics
(Section 6.3).
2. The weak-selection limit for juvenile migration is panmixia (Lou and Nagylaki
2008).
3. For hard selection (3.10), the dynamics in the Levene model becomes
p0i = pi
wi
w̄
(i ∈ I) , where wi =
X
cα wi,α ,
(5.5)
α
which is again equivalent to the panmictic dynamics with fitnesses averaged over all demes.
Therefore, no conceptually new behavior occurs. With two alleles, there exists a protected
polymorphism if and only if the averaged fitnesses display heterozygous advantage. Interestingly, it is not true that the condition for protection of an allele under hard selection is
always more stringent than under soft selection, although under a range of assumptions
this is the case (see Nagylaki 1992, Sect. 6.3).
5.1
General results about equilibria and dynamics
We define
w̃ = w̃(p) =
Y
w̄α (p)cα ,
(5.6)
α
which is the geometric mean of the mean fitnesses in single demes. Furthermore, we define
F (p) = ln w̃(p) =
X
cα ln w̄α (p) .
(5.7)
α
Both functions will play a crucial role in our study of the Levene model.
From
X
∂ w̄α
∂ X
=
pi pj wij,α = 2
pj wij,α = 2wi,α ,
∂pi
∂pi i,j
j
40
(5.8)
we obtain
p0i = pi
Because
P
i
X cα 1 ∂ w̄α
1 X ∂ ln w̄α
1 ∂F (p)
= pi
cα
= pi
.
w̄
2
∂p
2
∂p
2
∂p
α
i
i
i
α
α
p0i = 1, we can write (5.9) in the form
∂ w̃(p) X ∂ w̃(p)
∂F (p) X ∂F (p)
0
pj
= pi
pj
.
pi = pi
∂pi
∂pj
∂pi
∂pj
j
j
(5.9)
(5.10)
A simple exercise using Lagrange multipliers shows that the equilibria of (5.10), hence
of (5.3), are exactly the critical points of w̃, or F . Our aim is to prove the following
theorem.
Theorem 5.2 (Li 1955; Cannings 1971; Nagylaki 1992, Sect. 6.3). (a) Geometric-mean
fitness satisfies ∆w̃(p) ≥ 0 for every p ∈ SI , and ∆w̃(p) = 0 if and only if p is an
equilibrium of (5.3). The same conclusion holds for F .
(b) The set Λ of equilibria is globally attracting, i.e., p(t) → Λ as t → ∞. If every
point in Λ is isolated, as is generic7 , then p(t) converges to some p̂ ∈ Λ. Generically, p(t)
converges to a local maximum of w̃.
This is an important theoretical result because it states that in the Levene model no
complex dynamical behavior, such as limit cycles or chaos, can occur. One immediate
consequence of this theorem is
Corollary 5.3. Assume the Levene model with two alleles. If there exists a protected
polymorphism, then all trajectories converge to an internal equilibrium point.
Outline of the proof of Theorem 5.2. (a) We assume cα ∈ Q for every α. Then there
exists n ∈ N such that W = w̃n is a homogeneous polynomial in p with nonnegative
coefficients. From
we infer
∂ w̃
∂W
= nw̃n−1
,
∂pi
∂pi
∂W
∂pi
X
j
∂W
∂ w̃
pj
=
∂pj
∂pi
X
j
pj
∂ w̃
.
∂pj
From the inequality of Baum and Eagon (1967), we infer immediately W (p0 ) > W (p)
unless p0 = p. Therefore, W and w̃ are (strict) Lyapunov functions. By a straightforward
but tedious approximation argument, which uses compactness of SI , it follows that w̃ is
7
We call a property generic if it holds for almost all parameter combinations and, if applicable, for
almost all initial conditions
41
also a Lyapunov function if cα ∈ R (see Nagylaki 1992, Sect. 6.3). This proves the first
assertion in (a). The second follows because the logarithm is strictly monotone increasing.
(b) The first statement is an immediate consequence of LaSalle’s invariance principle
(LaSalle 1976, p. 10). The second is obvious, and the third follows because w̃ is nondecreasing. (Note, however, that convergence to a maximum holds only generically because
some trajectories may converge to saddle points. Moreover, because w̃ may also have
minima and saddle points, the globally attracting set Λ may not be stable; only a subset
of Λ is stable.)
Next, we derive a useful criterion for the existence of a unique stable equilibrium.
Lemma 5.4. If w̄α is concave for every α, then F is concave.
Proof. Because the logarithm is strictly monotone increasing and concave, a simple estimate shows that ln w̄α is concave. Hence, F is concave.
Theorem 5.5 (Nagylaki and Lou 2001). If F is concave, there exists exactly one stable
equilibrium (point or manifold) and it is globally attracting. If there exists an internal
equilibrium, it is the global attractor.
Proof. Concavity and the fact that ∆F ≥ 0 imply that if an internal equilibrium exists,
it must be the global attractor.
Suppose now there exist two stable equilibrium points on the boundary of the simplex
SI . Because F is concave, F is constant on the line that joins them. If that line is on the
boundary of SI , then these two equilibrium points are elements of the same manifold of
equilibria (on the boundary of SI ). If the line connecting them is an internal equilibrium,
then the case treated above applies.
If dominance is absent in every deme, there exist constants vi,α such that
wij,α = vi,α + vj,α
(5.11)
for every i, j ∈ I and α ∈ G, whence (3.2) yields
wi,α = vi,α + v̄α
and w̄α = 2v̄α ,
(5.12)
where
v̄α =
X
i
42
vi,α pi .
(5.13)
A simple calculation shows that without dominance the dynamics (5.3) simplifies to
X vi,α 0
1
pi = 2 p i 1 +
cα
(i ∈ I) .
(5.14)
v̄
α
α
Fitnesses are called multiplicative if there exist constants vi,α such that
wij,α = vi,α vj,α
(5.15)
for every i, j ∈ I and α ∈ G. Then (3.2) yields
wi,α = vi,α v̄α
and w̄α = v̄α2 .
(5.16)
With multiplicative fitnesses, one obtains the haploid Levene model, i.e.,
p0i = pi
X
cα vi,α /v̄α
(i ∈ I) .
(5.17)
α
Theorem 5.6 (Nagylaki and Lou 2001). The function F is concave in the following cases:
(a) In every deme there is no dominance.
(b) In every deme fitnesses are multiplicative.
(c) In every deme a globally attracting internal equilibrium exists without migration
(i.e., with the dynamics p0i,α = pi,α wi,α /w̄α ).
Therefore, the conclusions of Theorem 5.5 apply in each of the cases.
Proof. (a) Without dominance, w̄α = 2v̄α is linear in every deme.
(b) With multiplicative fitnesses, ln w̄α = 2 ln v̄α , which is concave because v̄α is linear.
(c) If there is a globally attracting internal equilibrium in deme α when it is isolated,
then the fact that ∆w̄α ≥ 0 implies that the quadratic w̄α must be concave.
In all three cases, the assertion follows from Lemma 5.4.
Remark 5.7. Theorems 5.5 and 5.6 hold for hard selection if we replace F by w̄. For
haploids, Strobeck (1979) proved uniqueness and asymptotic stability of an internal equilibrium.
Nagylaki and Lou (2006b) provide sufficient conditions for the nonexistence of an
internal equilibrium.
The following theorem shows that a globally stable internal equilibrium can be maintained on an open set of parameters, even in the absence of overdominance. Clearly,
43
this requires that there are at least two demes; cf. Section 2.3. We say there is partial
dominance in every deme if
wii,α < wij,α < wjj,α
or wjj,α < wij,α < wii,α
(5.18)
holds for every pair i, j ∈ I with i 6= j and for every α ∈ Γ.
Theorem 5.8. Assume an arbitrary number of alleles and Γ ≥ 2 demes. Then there
exists a nonempty open set of fitness parameters exhibiting partial dominance such that
for every parameter combination in this set, there is a unique, internal, asymptotically
stable equilibrium point of the dynamics (5.3). This equilibrium is globally asymptotically
stable.
This theorem arises for the special case of a single locus in Result 5.1 in Bürger (2010).
The latter is an immediate consequence of Theorem 3.1 in Bürger (2010) and of Theorem
2.2, Remark 2.3(iii), and Remark 2.4 in Bürger (2009b); cf. Theorem 7.4 below. The
proof shows that the maintenance of a stable internal equilibrium requires a certain form
of spatially averaged overdominance that, with intermediate dominance, can be achieved
only if the direction of selection and the degree of dominance vary among demes.
5.2
No dominance
Throughout this subsection, we assume no dominance, i.e., (5.11). For this important
special case more detailed results can be proved than for general (intermediate) dominance. Nevertheless, internal equilibrium solutions can be determined explicitly only
under specific assumptions (e.g., Theorem 4.2 in Nagylaki and Lou 2006b). An interesting question to ask is: What determines the number of alleles that can be maintained at
an equilibrium? If there is no dominance, there is a simple answer.
Theorem 5.9 (Nagylaki and Lou 2001). Without dominance, the number of demes is a
generic upper bound on the number of alleles present at equilibrium. Any neutral deme
should not be counted in this bound.
Proof. From (5.14) we conclude that, at an internal equilibrium p̂,
X vi,α
cα
= 1 (i ∈ I)
v̄ˆα
α
(5.19)
holds. The substitution
xα = cα /v̄ˆα , where v̄ˆα =
X
i
44
vi,α p̂i ,
(5.20)
linearizes (5.19):
X
vi,α xα = 1 (i ∈ I) .
(5.21)
α
This is a system of I inhomogeneous linear equations for the Γ unknowns xα . Therefore,
a solution exists generically only if I ≤ Γ. (Note that even if there exists a solution (xα ),
it does not necessarily give rise to a solution p̂ ∈ SI .)
The statement about neutral demes follows because in a neutral deme vi,α = v̄α = 1
for every i, hence xα = cα holds.
Remark 5.10. If in Theorem 5.9 general intermediate dominance is admitted, then there
exists an open set of parameters for which any number of alleles can be maintained at an
asymptotically stable equilibrium. Theorem 7.2 provides a much more general result.
Remark 5.11. Theorem 5.9 holds for hard selection. A slight modification of the proof
shows that it holds also for multiplicative fitnesses. In fact, Strobeck (1979) proved an
analog of Theorem 5.9 for haploid species.
Example 5.12 (Nagylaki and Lou 2001). The upper bound established in Theorem 5.9
can be assumed if Γ = I. Let vi,α = ui δiα , where ui > 0 and δiα is the Kronecker delta.
This means that allele Ai has fitness ui in deme i and fitness 0 elsewhere. Hence, every
allele is the best in one deme. Then (5.14) simplifies to
ui
0
1
pi = 2 pi 1 +
= 12 pi (1 + ci /pi ) = 12 (pi + ci ) .
v̄i
(5.22)
The solution is pi (t) = ci + (pi (0) − ci )( 12 )t → ci as t → ∞.
In the formulation of Theorem 5.9, the word ‘generic’ is essential. If Γ and I are
arbitrary and one assumes
wij,α = 1 + rij gα
(5.23)
for (sufficiently small) constants rij and gα , it can be shown that an internal (hence globally
attracting) manifold of equilibria exists for an open set of parameter combinations. This
holds also for the additive case, when rij = si + sj (Nagylaki and Lou 2001).
Another interesting problem is to determine conditions when a specific allele will go
to fixation. A simple and intuitive result is the following:
Theorem 5.13 (Nagylaki and Lou 2006b). Suppose there exists i ∈ I such that
X vj,α
cα
<1
vi,α
α
for every j 6= i. Then pi (t) → 1 as t → ∞.
45
(5.24)
Figure 5.1: Phase portrait for the Levene model in Example 5.14 (Figure from Nagylaki and
Lou 2001).
A proof as well as additional results on the loss or fixation of alleles may be found
in Nagylaki and Lou (2006b). The following example is based on a nice application of
Theorem 5.6 and shows that migration may eliminate genetic variability.
Example 5.14 (Nagylaki and Lou 2001). We suppose two demes of equal size (c1 = c2 =
1
)
2
and three alleles without dominance. The alleles A1 and A2 have extreme fitnesses
and A3 is intermediate in both demes. More precisely, we assume v1,1 = 1, v2,1 = 0,
v3,1 = u, and v1,2 = 0, v2,2 = 1, v3,2 = u, where 0 < u < 1. Without migration, A1 is
ultimately fixed in deme 1 and A2 is fixed in deme 2. We now establish that the Levene
model evolves as sketched in Figure 5.1, i.e.,
(
( 21 , 21 , 0) if 0 < u < 12 ,
(5.25)
p(t) →
(0, 0, 1) if 12 ≤ u < 1 .
Because there is no dominance, Theorem 5.6 informs us that there can be only one stable
equilibrium. Therefore, it is sufficient to establish asymptotic stability of the limiting
equilibrium, which we leave as an easy exercise.
Following Nagylaki (2009a), we say there is deme-independent degree of intermediate
dominance (DIDID) if
wij,α = ϑij wii,α + ϑji wjj,α
(5.26a)
holds for constants ϑij such that
0 ≤ ϑij ≤ 1 and ϑji = 1 − ϑij
for every α and every pair i, j. In particular, ϑii = 21 .
46
(5.26b)
Obviously, DIDID covers complete dominance or recessiveness (ϑij = 0 or ϑij = 1 if
i 6= j), and no dominance (ϑij = 12 ), but not multiplicativity. We also note that DIDID
includes the biologically important case of absence of genotype-by-environment interaction. In general, F is not concave under DIDID. However, Nagylaki (2009a, Theorem 3.2)
proved that with DIDID the evolution of p(t) is qualitatively the same as without dominance. Therefore, the conclusion of Theorem 5.5 holds, i.e., under DIDID there exists
exactly one stable equilibrium (point or manifold) and it is globally attracting. If there
exists an internal equilibrium, it is the global attractor. Moreover, the number of demes
is a generic upper bound on the number of alleles that can segregate at an equilibrium
(generalization of Theorem 5.9). This contrasts sharply with Theorem 5.8. Finally, the
condition for loss of an allele in Theorem 5.13 has a simple generalization to DIDID. For
proofs and further results about DIDID consult Nagylaki (2009a).
Remark 5.15. For general migration, the dynamics under DIDID may differ qualitatively
from that under no dominance. For instance, in a diallelic model with one way migration,
two asymptotically stable equilibria may coexist if there is DIDID (Nagylaki 2009a).
Moreover, in two triallelic demes, a globally asymptotically stable internal equilibrium
exists for an open set of parameters (Peischl 2010). Therefore, with arbitrary migration
and DIDID, the number of alleles maintained at a stable equilibrium can be higher than
the number of demes. This is not the case in the absence of dominance when the number
of demes is a generic upper bound (Theorem 6.1).
5.3
Two alleles with dominance
Here, we specialize to two alleles but admit arbitrary dominance. We assume that fitnesses
are given by (4.1). Our first result yields a further class of examples when a unique stable
equilibrium exists.
Theorem 5.16 (Bürger 2009c). For every α, let the fitnesses satisfy
xα yα ≤ 1 + (1 − yα )2 and yα ≤ 1
(5.27a)
xα yα ≤ 1 + (1 − xα )2 and xα ≤ 1 .
(5.27b)
or
Then F is concave on [0, 1]. Hence, there exists at most one internal equilibrium. If an internal equilibrium exists, it is globally asymptotically stable. If a monomorphic equilibrium
is stable, then it is globally asymptotically stable.
47
In the proof it is shown that ln w̄α is concave if and only if (5.27) is fulfilled. Then the
conclusion follows from Theorem 5.5.
Theorem 5.16 shows that the protection conditions (4.29) imply the existence of a
globally asymptotically stable internal equilibrium if (5.27) holds. It generalizes Result I
in Karlin (1977), who proved uniqueness of an internal equilibrium and global convergence
under the assumption of submultiplicative fitnesses (4.47). His proof is based on a different
method. Submultiplicative fitnesses, hence fitnesses satisfying (5.27), include a number
of important cases: multiplicative fitnesses (hence, selection on haploids), no dominance,
partial or complete dominance of the fitter allele, and overdominance.
Example 5.17. If there are two niches, no dominance, and fitnesses are given by 1 + sα ,
1, and 1 − sα (s1 s2 < 0), then (4.15) informs us that there is protected polymorphism if
|κ| < 1 ,
where κ =
c1 c2
+ .
s2 s1
(5.28)
By Theorem 5.6 or 5.16 there is a unique, asymptotically stable equilibrium p̂. A simple
calculation yields p̂ = (1 − κ)/2.
Example 5.18. If there are two niches and multiplicative fitnesses given by 1/(1 − sα ),
1, and 1 − sα (s1 s2 < 0), then there is a protected polymorphism if
0 < 1 − κ < 1,
(5.29)
where κ is as above. The unique, asymptotically stable equilibrium is given by p̂ = 1 − κ.
A partial converse to Theorem 5.16 is the following:
Theorem 5.19 (Karlin 1977, Result IA). Suppose
xα yα > 1 + max{(xα − 1)2 , (yα − 1)2 }
for every α ,
(5.30)
and there exists at least one internal equilibrium. Then both monomorphic equilibria are
asymptotically stable and the internal equilibrium is unique.
Figure 5.2 displays the regions in which submultiplicativity, (5.27), or (5.30) hold.
Table 5.1 presents numerical results which demonstrate that stronger selection, submultiplicativity, and a higher number of demes all facilitate protected polymorphism in the
Levene model. These findings agree with intuition. For instance, if an allele has sufficiently low fitness in just one deme, i.e., cα /yα > 1, the other allele is protected.
48
Figure 5.2: In the light blue regions fitnesses are submultiplicative, in the dark blue and light
blue regions they satisfy (5.27). In the blue region(s), there exists a single stable equilibrium. In
the purple region, fitnesses satisfy (5.30). There, both boundary equilibria are stable provided
an internal equilibrium exists. In the light blue square there is overdominance, in the light
purple square there is underdominance.
So far, we derived sufficient conditions for a unique (internal) equilibrium, but we have
not yet considered the question of how many (stable) internal equilibria can coexist. This
turns out to be a difficult question which was solved only recently for diallelic loci.
With fitnesses given by (4.1), a simple calculation shows that the dynamics (5.3) can
be written as
∆p = p(1 − p)
X
α
cα
1 − yα + p(xα + yα − 2)
.
xα p2 + 2p(1 − p) + yα (1 − p)2
(5.31)
Expressing this with a common denominator, we see that the internal equilibria are the
solutions of a polynomial in p of degree 2Γ − 1. Thus, in principle, there can be up to
2Γ − 1 internal equilibria. For two demes, Karlin (1977) provided a construction principle
for obtaining three internal equilibria. It requires quite extreme fitness differences and
that in both demes the less fit allele is clsoe to dominant.
By a clever procedure, Novak (2011) proved the following result:
Theorem 5.20. The diallelic Levene model allows for any number j ∈ {1, 2, . . . , 2Γ − 1}
of hyperbolic internal equilibria with any feasible stability configuration.
His numerical results, which admit arbitrary dominance, show that the proportion of
49
Γ
2
3
4
10
weak selection
intdom submult
0.094
0.185
0.124
0.262
0.138
0.310
0.158
0.468
moderate
intdom
0.115
0.162
0.190
0.270
selection
submult
0.271
0.416
0.512
0.788
strong selection
intdom submult
0.220
0.332
0.357
0.516
0.460
0.636
0.787
0.917
Table 5.1: Proportion of protected polymorphism in the Levene model with intermediate dominance. Fitnesses of A1 A1 , A1 A2 , A2 A2 are parameterized as 1+sα , 1+hα sα , 1−sα . The data for
weak selection are generated for the weak-selection limit (Remark 5.1). P
Then there is a protected
P
polymorphism if and only if there is average overdominance, i.e., if α cα sα hα > | α cα sα |
holds. For moderate or strong selection, the conditions (4.29) were evaluated. In all cases,
106 parameter combinations satisfying sα ∈ (−s, s), hα ∈ (−1, 1), cα ∈ (−1, 1) were randomly
(uniformly) chosen, and the values cα were normalized. For weak or strong selection, s = 1;
for moderate selection, s = 0.2. For submultiplicative fitnesses (columns ‘submult’), the 106
parameter combinations satisfy the additional constraint (4.47), which is reformulated as a condition for hα . The columns ‘intdom’ contain the data for general intermediate dominance. The
proportion of submultiplicative parameter combinations among all parameter combinations with
intermediate dominance is (1 − 21 ln 2)Γ ≈ 0.653Γ .
parameter space supporting more than Γ equilibria becomes extremely small if Γ > 2.
6
Multiple alleles and arbitrary migration
Because for multiple alleles and arbitrary migration few general results are available, we
focus on three limiting cases that are biologically important and amenable to mathematical analysis. These are weak selection and weak migration, weak migration (relative to
selection), and weak selection (relative to migration). The first leads to the continuoustime migration model, the second to the so-called weak-migration limit, and the third to
the strong-migration limit. The latter two are based on a separation of time scales. With
the help of perturbation theory, results about existence and stability of equilibria, but
also about global convergence, can be derived for an open subset of parameters of the full
model. In addition, we report results about the case of no dominance and about uniform
selection, i.e., when selection is the same in every deme. We start with no dominance.
6.1
No dominance
The following generalizes Theorem 5.9 for the Levene model. It holds for an arbitrary
(constant) backward migration matrix.
50
Theorem 6.1 (Nagylaki and Lou 2001, Theorem 2.4). Without dominance, the number
of demes is a generic upper bound on the number of alleles that can be maintained at any
equilibrium.
This theorem also holds for hard selection. As shown by Peischl (2010), it can not
be extended to DIDID; cf. Remark 5.15. An example showing that the upper bound can
be achieved if I = Γ is obtained by setting vi,α = ui δiα , where ui > 0. The internal
equilibrium can be written as p̂ = 21 (I − 21 M )−1 M , and convergence is geometric (at a
rate ≤ 21 ).
As we shall see below, with dominance the number of alleles that can be maintained
at equilibrium may depend on the strength and pattern of migration.
6.2
Migration and selection in continuous time
Following Nagylaki and Lou (2007), we assume that both selection and migration are
weak and approximate the discrete migration-selection dynamics (3.5) by a differential
equation which is easier accessible. Accordingly, let
wij,α = 1 + rij,α
and m̃αβ = δαβ + µ̃αβ ,
(6.1)
where rij,α and µ̃αβ are fixed for every i, j ∈ I and every α, β ∈ G, and > 0 is sufficiently
small. From (3.2) we deduce
wi,α = 1 + ri,α
and w̄α = 1 + r̄α ,
(6.2a)
where
ri,α =
X
rij,α pj,α
and r̄α =
j
X
rij,α pi,α pj,α .
(6.2b)
i,j
To approximate the backward migration matrix M , note that (3.10) and (6.2a) imply
that, for both soft and hard selection,
c∗α = cα + O()
(6.3)
as → 0. Substituting (6.1) and (6.3) into (3.8) leads to
mαβ = δαβ + µαβ + O(2 )
(6.4)
as → 0, where
µαβ
1
=
cα
!
cβ µ̃βα − δαβ
X
γ
51
cγ µ̃γα
.
(6.5)
Because M̃ is stochastic, we obtain for every α ∈ Γ,
µ̃αβ ≥ 0 for every β 6= α and
X
µ̃αβ = 0 .
(6.6)
β
As a simple consequence, µαβ shares the same properties.
The final step in our derivation is to rescale time as in Sect. 2.4 by setting t = bτ /c
and πi,α (τ ) = pi,α (t). Inserting all this into the difference equations (3.5) and expanding
yields
πi,α (τ + ) = πi,α {1 + [ri,α (π·,α ) − r̄α (π·,α )]} + X
µαβ πi,β + O(2 )
(6.7)
β
as → 0, where π·,α = (π1,α , . . . , πI,α )> ∈ SI . Rearranging and letting → 0, we arrive at
dπi,α X
=
µαβ πi,β + πi,α [ri,α (π·,α ) − r̄α (π·,α )] .
dτ
β
(6.8)
Absorbing into the migration rates and selection coefficients and returning to p(t), we
obtain the slow-evolution approximation of (3.5),
ṗi,α =
X
µαβ pi,β + pi,α [ri,α (p·,α ) − r̄α (p·,α )] .
(6.9)
β
In contrast to the discrete-time dynamics (3.5), here the migration and selection terms are
decoupled. This is a general feature of many other slow-evolution limits (such as mutation
and selection, or selection, recombination and migration). Because of the decoupling of
the selection and migration terms, the analysis of explicit models is often facilitated.
With multiple alleles, there are no general results on the dynamics of (6.9). For two
alleles, we set pα = p1,α and write (6.9) in the form
ṗα =
X
µαβ pβ + ϕα (pα ) .
(6.10)
β
Since µαβ ≥ 0 whenever α 6= β, the system (6.10) is quasimonotone or cooperative, i.e.,
∂ ṗα /∂pβ ≥ 0 if α 6= β. As a consequence, (6.10) cannot have an exponentially stable limit
cycle. However, Akin (personal communication) has proved for three diallelic demes that
a Hopf bifurcation can produce unstable limit cycles. This precludes global convergence,
though not generic convergence. If Γ = 2, then every trajectory converges (Hirsch 1982;
Hadeler and Glas 1983; see also Hofbauer and Sigmund 1998, p. 28), as is the case in the
discrete-time model (Remark 4.15).
52
Example 6.2. Eyland (1971) provided a global analysis of (6.9) for the special case of
two diallelic demes without dominance. As in Sect. 4.2, we assume that the fitnesses of
A1 A1 , A1 A2 , and A2 A2 in deme α are 1 + sα , 1, and 1 − sα , respectively, where sα 6= 0
(α = 1, 2). Moreover, we set µ1 = µ12 > 0, µ2 = µ21 > 0, and write pα for the frequency
of A1 in deme α. Then (6.9) becomes
ṗ1 = µ1 (p2 − p1 ) + s1 p1 (1 − p1 ) ,
(6.11a)
ṗ2 = µ2 (p1 − p2 ) + s2 p2 (1 − p2 ) .
(6.11b)
The equilibria can be calculated explicitly. At equilibrium, p1 = 0 if and only if p2 = 0,
and p1 = 1 if and only if p2 = 1. In addition, there may be an internal equilibrium point.
We set
σα =
µα
,
sα
κ = σ1 + σ2 ,
(6.12)
and
B = (1 − 4σ1 σ2 )1/2 .
(6.13)
The internal equilibrium exists if and only if s1 s2 < 0 and |κ| < 1; cf. (4.15). If s2 < 0 < s1 ,
it is given by
p̂1 = 21 (1 + B) − σ1
and p̂2 = 12 (1 − B) − σ2 .
(6.14)
It is straightforward to determine the local stability properties of the three possible
equilibria. Gobal asymptotic stability follows from the results cited above about quasimonotone systems. Let p = (p1 , p2 )> . Then allele A1 is eliminated in the region Ω0 in
Figure 4.2, i.e., p(t) → (0, 0) as t → ∞, whereas A1 is ultimately fixed in the region Ω1 .
In Ω+ , p(t) converges globally to the internal equilibrium point p̂ given by (6.14).
For the discrete-time dynamics (3.5) such a detailed analysis is not available. However,
for important special cases, results about existence, uniqueness, and stability of equilibria
were derived by Karlin and Campbell (1980). Some of them are treated in Section 4.5.
Finally, we present sufficient conditions for global loss of an allele. In discrete time,
such conditions are available only for the Levene model. For (6.9), however, general
conditions were derived by Nagylaki and Lou (2007). Suppose that there exists i ∈ I and
constants γij such that
γij ≥ 0 ,
γii = 0 ,
X
γij = 1 ,
(6.15a)
j
and
X
γij rjk,α > rik,α
j
53
(6.15b)
for every α ∈ G and every k ∈ I.
Theorem 6.3 (Nagylaki and Lou 2007, Theorem 3.5 and Remark 3.7). If the matrix
(µαβ ) is irreducible and the conditions (6.15) are satisfied, then pi (t) → 0 as t → ∞
whenever pi (0) > 0 and pj (0) > 0 for every j such that γij > 0.
If there is no dominance, constants si,α exist such that rij,α = si,α + sj,α for every i, j, α.
Then condition (6.15b) simplifies to
X
γij sj,α > si,α ,
(6.16)
j
and Theorem 6.3 applies.
To highlight one of the biological implications, we follow Remark 3.12 in Nagylaki and
Lou (2007) and assume γi1 > 0, γiI > 0, and γij = 0 for j = 2, . . . , I − 1. Then (6.16)
becomes
γi1 s1,α + γiI sI,α > si,α
(6.17)
for every α, and the theorem shows that pi (t) → 0 as t → ∞ for i = 2, . . . , I − 1. If, in
addition to (6.17), we assume that
min(s1,α , sI,α ) < si,α < max(s1,α , sI,α )
(6.18)
for i = 2, . . . , I − 1 and every α, then every allele Ai with 1 < i < I is intermediate in
every deme, and A1 and AI are extreme. Thus, Theorem 6.3 implies that all intermediate
alleles are eliminated. This conclusion can be interpreted as the elimination of generalists by specialists, and it can yield the increasing phenotypic differentiation required for
parapatric speciation (cf. Lou and Nagylaki 2002 for an analogous result and discussion
in the context of diffusion models). If s1,α − sI,α changes sign among demes, so that every
allele is the fittest in some deme(s) and the least fit in the other(s), then both alleles may
be maintained.
An other application of Theorem (6.3) is the following. If in every deme the homozygotes have the same fitness order and there is strict heterozygote intermediacy, then the
allele with the greatest homozygous fitness is ultimately fixed (Remark 3.20 in Nagylaki
and Lou 2007).
Remark 6.4. The slow-evolution approximation of the exact juvenile-migration model
is also (6.9) (Nagylaki and Lou 2008, Nagylaki 1992, pp. 143-144).
54
6.3
Weak migration
If migration is sufficiently weak relative to selection, properties of the dynamics can be
inferred by perturbation techniques from the well-understood case of a finite number of
isolated demes in which there is selection and random mating.
To study weak migration, we assume
mαβ = δαβ + µαβ ,
(6.19)
where µαβ is fixed for every α, β, and > 0 is sufficiently small. Because M is stochastic,
we obtain for every α ∈ Γ (cf. Section 6.2)
X
µαβ ≥ 0 for every β 6= α and
µαβ = 0 .
(6.20)
β
If there is no migration ( = 0), the dynamics in each deme reduces to the pure selection
dynamics
wi,α
.
(6.21)
w̄α
Because (6.21) is defined on the Cartesian product SΓI , it may exhibit a richer equilibrium
p0i,α = pi,α
and stability structure than the panmictic selection dynamics (2.6). This was already
illustrated by Example 4.5, in which two asymptotically stable internal equilibria may
coexist. By Theorem 2.2, such an equilibrium configuration does not occur for (2.6).
The following is a central perturbation result that has a number of important consequences. Among others, it excludes complex dynamics in the full system (3.5) provided
migration is sufficiently weak. We note that hyperbolicity is a generic property for (6.21)
(Appendix A in Nagylaki et al. 1999), and an internal equilibrium is hyperbolic if and
only if it is isolated (Lemma 3.2 in Nagylaki and Lou 2006a).
Theorem 6.5 (Nagylaki and Lou 2007, Theorem 4.1). Suppose that every equilibrium of
(6.21) is hyperbolic, that (6.19) holds, and that > 0 is sufficiently small.
(a) The set of equilibria Σ0 ⊂ SΓI of (6.21) contains only isolated points, as does the
set of equilibria Σ ⊂ SΓI of (3.5). As → 0, each equilibrium in Σ converges to the
corresponding equilibrium in Σ0 .
(b) In the neighborhood of each asymptotically stable equilibrium in Σ0 , there exists
exactly one equilibrium in Σ , and it is asymptotically stable. In the neighborhood of each
unstable internal equilibrium in Σ0 , there exists exactly one equilibrium in Σ , and it is
unstable. In the neighborhood of each unstable boundary equilibrium in Σ0 , there exists at
most one equilibrium in Σ , and if it exists, it is unstable.
(c) Every solution p(t) of (3.5) converges to one of the equilibrium points in Σ .
55
The perturbation results in (a) and (b) are essentially due to Karlin and McGregor
(1972a,b). The proof of (c), which also yields (a) and (b), is a simplification of that of
Theorem 2.3 in Nagylaki et al. (1999) and is based on quite deep results. To outline the
proof, we need some preparation.
We consider a family of maps (difference equations) that depends on a parameter ,
x0 = f (x, ) ,
(6.22)
where x ∈ X ⊆ Rn (X compact and convex) and ∈ E ⊆ Rk (E open). We assume that x̂
is a fixed point of (6.22) if = 0 and that the Jacobian f 0 (x̂, 0 ) of f evaluated at (x̂, 0 )
exists and is continuous. Furthermore, we posit that x̂ is a hyperbolic equilibrium. If we
define the function
F (x, ) = f (x, ) − x ,
(6.23)
then F (x̂, 0 ) = 0 and the Jacobian F 0 (x̂, 0 ) is continuous and nonsingular (by hyperbolicity of x̂). Therefore, the implicit function theorem shows that there exists an open
neighborhood U of 0 and a function φ : U → X such that φ(0 ) = x̂ and F (φ(), ) = 0
for ∈ U, hence
f (φ(), ) = φ() .
(6.24)
Hence, for every ∈ U, i.e., for close to 0 , (6.22) has a uniquely determined fixed point
x̂ = φ() close to x̂.
With the help of the Hartman-Grobman theorem, it can be shown that the stability
properties of the perturbed fixed points x̂ are the same as those of the unperturbed,
x̂. The reason is that if an equilibrium is hyperbolic, this property persists under small
perturbations (the Jacobian changes continuously if parameters change continuously); see
also Theorem 4.4 in Karlin and McGegor (1972b). The extension of this argument to
finitely many hyperbolic equilibria is evident.
Although hyperbolic equilibria change continuously under small perturbations, limit
sets of trajectories do not: they can explode. Thus, perturbations could introduce ‘new’
limit sets away from the hyperbolic equilibria. What has good properties under perturbations is the set of chain-recurrent points introduced by Conley (1978).
Let X be a compact set with metric d and let f : X → X be a continuous map. A
point x ∈ X is called chain recurrent (with respect to f ) if, for every δ > 0, there exists
a finite sequence x0 = x, x1 , . . . , xr−1 , xr = x (often called a δ-pseudo-orbit) such that
d(f (xm ), xm+1 ) < δ for m = 0, 1, . . . , r − 1. The set of chain-recurrent points contains the
limit sets of all orbits and behaves well under perturbations (Akin 1993, p. 244).
56
Outline of the proof of Theorem 6.5. (a) and (b) follow from the implicit function theorem and the Hartman-Grobman theorem. That asymptotically stable boundary equilibria
remain in the state space follows from Brouwer’s fixed point theorem (for details, see Karlin and McGregor 1972, especially Theorem 4.4).
(c) Let
F = {p ∈ SΓI : pi,α (wi,α − w̄α ) = 0 ∀i ∈ I, ∀α ∈ Γ}
(6.25)
denote the set of equilibria of (6.21). Then, within in each deme, ∆w̄α ≥ 0 with equality
only at equilibrium; cf. (2.13). Hence,
w̄(p) =
X
w̄α (p·,α )
(6.26)
α
satisfies ∆w̄(p) ≥ 0 with ∆w̄(p) = 0 if and only if p ∈ F. Because, on F, w̄ takes only
finitely many values, Theorem 3.14 in Akin (1993) implies that the chain-recurrent points
of (6.21) are exactly the equilibria.
Now we can follow the proof of Theorem 2.3 in Nagylaki et al. (1999) almost verbally.
Because the set of chain-recurrent points consists only of hyperbolic equilibria, this is
also true for small C 1 perturbations of the dynamics (Akin 1993, p. 244). Indeed, as an
immediate consequence of the definition of chain recurrence, it follows that the chainrecurrent set of (6.21) changes in an upper semicontinuous way with . In particular,
the chain-recurrent set for > 0 is contained in the union of the δ-neighborhoods of
the equilibria for = 0, with δ → 0 for → 0. By the implicit function theorem
and the openness of hyperbolicity (Hartman-Grobman theorem), if > 0 is small, then
the maximal invariant sets in those neighborhoods are hyperbolic equilibria. Hence, for
small , the chain-recurrent set consists only of finitely many equilibria, which implies
convergence of all trajectories.
Remark 6.6. Boundary equilibria that are unstable in the absence of migration, can
disappear under weak migration because they may leave SΓI . A simple example is overdominance in two diallelic demes: the unstable zero-migration equilibria (p1,1 , p1,2 ) = (1, 0)
and (p1,1 , p1,2 ) = (0, 1) do not survive perturbation. Indeed, if > 0, the perturbation of
(1, 0) must have the form (1 − z1 , z2 ). If the fitnesses of the genotypes are as in (4.1)
and the migration rates are m12 = m1 and m21 = m2 , then straightforward calculations
show that this equilibrium is given by
z1 = −
x1 m1
1 − x1
and z2 = −
y 2 m2
,
1 − y2
which is not in [0, 1] × [0, 1] if there is overdominance, i.e., if xα < 1 and yα < 1.
57
Remark 6.7. In the absence of migration, mean fitness is monotone increasing in each
P
deme (except at the equilibria); see (2.13). Therefore, by continuity, ∆w̄(p) = α ∆w̄α (p·,α ) >
0 for sufficiently small if p is bounded away from the set F of equilibria. If p is close to
F, mean fitness may decrease. As an example assume two diallelic demes with overdominance and (stable) equilibria p̂·,1 and p̂·,2 with p̂·,1 6= p̂·,2 . Suppose that in some generation
p·,α = p̂·,α holds for α = 1, 2. Since w̄(p) is maximized at p̂, which, with migration, is an
equilibrium only if p̂·,1 = p̂·,2 , we see that w̄(p0 ) < w̄(p).
As another application of Theorem 6.5, we study the number of alleles that can be
maintained at an asymptotically stable equilibrium under weak migration. First we prove
a simple result for a single deme. For more general results, see Sect. 2 in Nagylaki and
Lou (2006a).
Proposition 6.8. Let Γ = 1 and assume that the alleles are ordered such that wii ≥
wi+1,i+1 for i = 1, . . . , I − 1. In addition, assume that w11 > w22 and there is intermediate
dominance, i.e.,
wii ≥ wij ≥ wjj
(6.27)
for every i and every j > i. Then allele A1 is fixed as t → ∞. The corresponding
equilibrium is globally asymptotically stable.
Proof. The assumptions imply w1i ≥ wii ≥ wIi for every i ∈ I and w11 > wI1 or wI1 > wII .
P
P
It follows that w1 = i w1i pi ≥ i wIi pi = wI and, if p1 6= 0 and pI 6= 0,
w1 > wI .
(6.28)
Therefore, we obtain
0
pI
p0
pI w I
pI
= I0 =
<
(6.29)
p1
p1
p1 w 1
p1
if p1 > 0 and pI > 0. Hence, pI (t) → 0 as t → ∞ provided pI < 1 and p1 > 0. Now we
can repeat this argument with p1 and pI−1 . Proceeding inductively, we obtain p1 (t) → 1
as t → ∞.
Theorem 6.9. Suppose that migration is sufficiently weak and there is partial dominance
in every deme (5.18).
(a) Generically, there is global convergence to an asymptotically stable equilibrium at
which at most Γ alleles are present. Thus, the number of demes is a generic upper bound
for the number of alleles that can be maintained at a stable equilibrium.
(b) If Γ ≤ I, then there is an open set of parameters such that Γ alleles are segregating
at a globally asymptotically stable equilibrium.
58
Proof. Corollary 6.8 shows that, in the absence of migration, one allele is fixed in every
deme. The parameter combinations satisfying the assumptions of the theorem clearly
form an open set of all possible parameter combinations. Therefore, without migration,
at most Γ alleles can be maintained at an asymptotically stable equilibrium, and this can
be achieved on an open set in the parameters space. Moreover, this equilibrium is globally
asymptotically stable. Therefore, Theorem 6.5 yields statement (a) for weak migration.
Clearly, the same set of alleles as without migration occurs at this equilibrium.
If Γ ≤ I, we still obtain an open set of parameters if we choose fitnesses such that in
each deme a different allele has the highest homozygous fitness. Hence, the upper bound
Γ can be achieved on an open set, and Theorem 6.5 yields statement (b).
Therefore, in contrast to the Levene model (Theorem 5.8), in which migration is strong,
for weak migration the number of alleles that can be maintained at a stable equilibrium
cannot exceed the number of demes.
6.4
Strong migration
If migration is much stronger than selection, we expect that rapid convergence to spatial
quasi-homogeneity occurs. After this short initial phase, evolution should be approximately panmictic with suitably averaged allele frequencies. Because in the absence of
selection, there exists a globally attracting manifold of equilibria, so that the dynamics
is not gradient like, the derivation of perturbation results is much more delicate than for
weak migration. Since the fundamental ideas in the proofs of the most relevant results
are essentially the same as if selection acts on many loci, we defer the analysis of strong
migration to Section 7.6, where the multilocus case is treated.
6.5
Uniform selection
Selection is called uniform if
wij,α = wij
for every i, j, and every α.
(6.30)
Under spatially uniform selection, one might expect that population structure leaves no
genetic traces. However, this is not always true as shown by Example 4.5 with underdominance and weak migration. Indeed, if we have two diallelic demes with the same
underdominant selection in both, then under weak migration there are nine equilibria,
four of which are asymptotically stable. These are the two monomorphic equilibria and
59
the two equilibria where each of the alleles is close to fixation in one deme and rare in
the other. Only three of the equilibria are uniform, i.e., have the same allele frequencies
in both demes. These are the two monomorphic equilibria and the ‘central’ equilibrium.
In the following we state sufficient conditions under which there is no genetic indication
of population structure. We call p̂ ∈ SΓI a uniform selection equilibrium if every p̂·,α is an
equilibrium of the pure selection dynamics (6.21) and p̂·,α = p̂·,β for every α, β.
Theorem 6.10 (Nagylaki and Lou 2007, Theorem 5.1). If p̂ ∈ SΓI is a uniform selection
equilibrium, then p̂ is an equilibrium of the (full) migration-selection dynamics (3.5), and
p̂ is either asymptotically stable for both (6.21) and (3.5), or unstable for both systems.
It can also be shown that the ultimate rate of convergence to equilibrium is determined
entirely by selection and is independent of migration.
Next, one may ask for the conditions when the solutions of (3.5) converge globally to
a uniform selection equilibrium. One may expect global convergence under migration if
the uniform selection equilibrium is globally asymptotically stable without migration. So
far, only weaker results could be proved.
Theorem 6.11 (Nagylaki and Lou 2007, Sections 5.2 and 5.3). Suppose there is a uniform selection equilibrium that is globally asymptotically stable in the absence of migration. Each of the following conditions implies global convergence of solutions to p̂ under
migration and selection.
(a) Migration is weak and, without migration, every equilibrium is hyperbolic.
(b) Migration is strong, M is ergodic, and every equilibrium of the strong-migration
limit is hyperbolic.
(c) The continuous-time model (6.9) applies, µαβ > 0 for every pair α, β with α 6= β,
and p̂ is internal in the absence of migration (hence, also with migration).
(d) The continuous-time model (6.9) applies, µαβ = µβα for every pair α, β, and p̂ is
internal.
7
Multilocus models
Since many phenotypic traits are determined by many genes, many loci are subject to
selection. If loci are on the same chromosome, especially if they are within a short physical
distance, they can not be treated independently. In order to understand the evolutionary
effects of selection on multiple loci, models have to be developed and studied that take
60
linkage and recombination into account. Because of the complexity of the general model,
useful analytical results can be obtained essentially only for limiting cases or under special
assumptions. Whereas the former can be sometimes extended using perturbation theory
to obtain general insight, the latter are mainly useful to study specific biological questions
or to demonstrate the kind of complexity that can arise.
Before developing the general model, we introduce the basic model describing the interaction of selection and recombination and point out some of its fundamental properties.
We begin by illustrating the effects of recombination in the simplest meaningful setting.
7.1
Recombination between two loci
We consider two loci, A and B, each with two alleles, A1 , A2 , and B1 , B2 . Therefore,
there are four possible gametes, A1 B1 , A1 B2 , A2 B1 , A2 B2 , and 16 diploid genotypes. If, as
usual, Ai Bj /Ak B` and Ak B` /Ai Bj are indistinguishable, then only 10 different unordered
genotypes remain.
Genes on different chromosomes are separated during meiosis with probability one half
(Mendel’s Principle of Independent Assortment). If the loci are on the same chromosome,
they may become separated by a recombination event (a crossover) between them. We
denote this recombination probability by r. The value of r usually depends on the distance
between the two loci along the chromosome. Loci with r = 0 are called completely linked
(and may be treated as a single locus), and loci with r = 21 are called unlinked. The
maximum value of r =
1
2
occurs for loci on different chromosomes, because then all four
gametes are produced with equal frequency
0≤r≤
1
.
4
Thus, the recombination rate satisfies
1
.
2
If, for instance, in the initial generation only the genotypes A1 B1 /A1 B1 and A2 B2 /A2 B2
are present, then in the next generation only these double homozygotes, as well as the
two double heterozygotes A1 B1 /A2 B2 and A1 B2 /A2 B1 will be present. After further
generations of random mating, all other genotypes will occur, but not immediately at
their equilibrium frequencies. The formation of gametic types other than A1 B1 or A2 B2
requires that recombination occurs between the two loci.
We denote the frequency of gamete Ai Bj by Pij and, at first, admit an arbitrary number
of alleles at each locus. Let the frequencies of the alleles Ai at the first locus be denoted
by pi and those of the alleles Bj at the second locus by qj . Then
pi =
X
Pij and qj =
j
X
i
61
Pij .
(7.1)
The allele frequencies are no longer sufficient to describe the genetic composition of the
population because, in general, they do not evolve independently. Linkage equilibrium
(LE) is defined as the state in which
Pij = pi qj
(7.2)
holds for every i and j. Otherwise the population is said to be in linkage disequilibrium
(LD). LD is equivalent to probabilistic dependence of allele frequencies between loci.
Given Pij , we want to find the gametic frequencies Pij0 in the next generation after
random mating. The derivation of the recursion equation is based on the following basic
fact of Mendelian genetics: an individual with genotype Ai Bj /Ak Bl produces gametes
of parental type if no recombination occurs (with probability 1 − r), and recombinant
gametes if recombination between occurs (with probability r). Therefore, the fraction of
gametes Ai Bj and Ak Bl is 12 (1 − r) each, and that of Ai Bl and Ak Bj is 12 r each. From
these considerations, we see that the frequency of gametes of type Ai Bj in generation t+1
produced without recombination is (1 − r)Pij , and that produced with recombination is
rpi qj because of random mating. Thus,
Pij0 = (1 − r)Pij + rpi qj .
(7.3)
This shows that the gene frequencies are conserved, but the gamete frequencies are not,
unless the population is in LE. Commonly, LD between alleles Ai and Bj is measured by
the parameter
Dij = Pij − pi qj .
(7.4)
The Dij are called linkage disequilibria. From (7.3) and (7.4) we infer
0
Dij
= (1 − r)Dij
(7.5)
Dij (t) = (1 − r)t Dij (0) .
(7.6)
and
Therefore, unless r = 0, linkage disequilibria decay at the geometric rate 1 − r and LE is
approached gradually without oscillation.
For two alleles at each locus, it is more convenient to label the frequencies of the
gametes A1 B1 , A1 B2 , A2 B1 , and A2 B2 by x1 , x2 , x3 , and x4 , respectively. A simple
calculation reveals that
D = x1 x4 − x2 x3
62
(7.7)
A1 B1
A2 B2
A2 B1
A1 B2
Figure 7.1: The tetrahedron represents the state space S4 of the two-locus two-allele
model. The vertices correspond to fixation of the labeled gamete, and frequencies are
measured by the (orthogonal) distance from the opposite boundary face. At the center
of the simplex all gametes have frequency 14 . The two-dimensional (red) surface is the
LE manifold, D = 0. The states of maximum LD, D = ± 14 , are the centers of the edges
connecting A1 B2 to A2 B1 and A1 B1 to A2 B2 .
satisfies
D = D11 = −D12 = −D21 = D22 .
(7.8)
Thus, the recurrence equations (7.3) for the gamete frequencies can be rewritten as
x0i = xi − ηi rD ,
i ∈ {1, 2, 3, 4} ,
(7.9)
where η1 = η4 = −η2 = −η3 .
The two-locus gametic frequencies are the elements of the simplex S4 and may be
represented geometrically by the points in a tetrahedron. The subset where D = 0 forms
a two-dimensional manifold and is called the linkage equilibrium, or Wright, manifold. It
is displayed in Figure 7.1.
If r > 0, (7.6) implies that all solutions of (7.9) converge to the LE manifold along
straight lines, because the allele frequencies x1 + x2 and x1 + x3 remain constant, where
sets such as x1 + x2 = const. represent planes in this geometric picture. The LE manifold
is invariant under the dynamics (7.9).
63
7.2
Two diallelic loci under selection
To introduce selection, we assume that viability selection acts on juveniles. Then recombination and random mating occurs. Since selection acts on diploid individuals, we assign
fitnesses to two-locus genotypes. We denote the fitness of genotype ij (i, j ∈ {1, 2, 3, 4})
by wij , where we assume wij = wji , because usually it does not matter which gamete
is paternally or maternally inherited. The marginal fitness of gamete i is defined by
P
P
wi = 4i=1 wij xj , and the mean fitness of the population is w̄ = 4i,j=1 wij xi xj . If we
assume, as is frequently the case, that there is no position effect, i.e., w14 = w23 , simple
considerations yield the selection-recombination dynamics (Lewontin and Kojima 1960):
x0i = xi
wi
w14
− ηi
rD ,
w̄
w̄
i ∈ {1, 2, 3, 4} .
(7.10)
This is a much more complicated dynamical system than either the pure selection dynamics (2.6) or the pure recombination dynamics (7.9), and has been studied extensively (for
a review, see Bürger 2000, Sects. II.2 and VI.2). In general, mean fitness may decrease
and is no longer maximized at an equilibrium.
In addition, the existence of stable limit cycles has been established for this discretetime model (Hastings 1981b, Hofbauer and Iooss 1984) as well as for the corresponding
continuous-time model (Akin 1979, 1982). Essentially, the demonstration of limit cycles
requires that selection coefficients and recombination rates are of similar magnitude.
There is a particularly important special case in which the dynamics is simple. This
(n)
is the case of no epistasis, or additive fitnesses. Then there are constants ui
(1)
(2)
wij = ui + uj
such that
for every i, j ∈ {1, 2, 3, 4} .
(7.11)
In the absence of epistasis, i.e., if (7.11) holds, mean fitness w̄ is a (strict) Lyapunov
function (Ewens 1969). In addition, a point p is an equilibrium point of (7.10) if and only
if it is both a selection equilibrium for each locus and it is in LE (Lyubich 1992, Nagylaki
et al. 1999). In particular, the equilibria are the critical points of mean fitness.
The reason for this increased complexity of two-locus (or multilocus) systems lies not
so much in the increased dimensionality but arises mainly from the fact that epistatic
selection generates nonrandom associations (LD) among the alleles at different loci. Recombination breaks up these associations to a certain extent but changes gamete frequencies in a complex way. Thus, there are different kinds of interacting nonlinearities arising
in the dynamical equations under selection and recombination.
64
7.3
The general model
We extend the migration-selection model of Section 3 by assuming that selection acts on
a finite number of recombining loci. The treatment follows Bürger (2009a), which was
inspired by Nagylaki (2009b). We consider a diploid population with discrete, nonoverlapping generations, in which the two sexes need not be distinguished. The population is
subdivided into Γ ≥ 1 panmictic colonies (demes) that exchange adult migrants independently of genotype. In each of the demes, selection acts through differential viabilities,
which are time and frequency independent. Mutation and random genetic drift are ignored.
(n)
The genetic system consists of L ≥ 1 loci and In ≥ 2 alleles, Ain (in = 1, . . . , In ),
at locus n. We use the multi-index i = (i1 , . . . , iL ) as an abbreviation for the gamete
(1)
(L)
Ai1 . . . AiL . We designate the set of all loci by L = {1, . . . , L}, the set of all alleles at
locus n by In = {1, . . . , In }, and the set of all gametes by I. The number of gametes is
Q
I = |I| = n In , the total number of genes (alleles at all loci) is I1 + · · · + IL . We use
letters i, j, ` ∈ I for gametes, k, n ∈ L for loci, and α, β ∈ G for demes. Sums or products
P
P
without ranges indicate summation over all admissible indices, e.g., n = n∈L .
Let pi,α = pi,α (t) represent the frequency of gamete i among zygotes in deme α in
generation t. We define the following column vectors:
pi = (pi,1 , . . . , pi,Γ )> ∈ RΓ ,
(7.12a)
p·,α = (p1,α , . . . , pI,α )> ∈ SI ,
> >
p = p>
∈ SΓI .
·,1 , . . . , p·,Γ
(7.12b)
(7.12c)
Here, pi , p·,α , and p signify the frequency of gamete i in each deme, the gamete frequencies
in deme α, and all gamete frequencies, respectively. We will use analogous notation for
other quantities, e.g. for Di,α .
(k)
The frequency of allele Aik among gametes in deme α is
(k)
pik ,α =
X
pi,α ,
(7.13)
i|ik
where the sum runs over all multi-indices i with the kth component fixed as ik . We write
>
(k)
(k)
(k)
pik = pik ,1 , . . . , pik ,Γ
∈ RΓ
(k)
for the vector of frequencies of allele Aik in each deme.
65
(7.14)
Let xij,α and wij,α denote the frequency and fitness of genotype ij in deme α, respectively. We designate the marginal fitness of gamete i in deme α and the mean fitness of
the population in deme α by
wi,α = wi,α (p·,α ) =
X
wij,α pj,α
(7.15a)
j
and
w̄α = w̄α (p·,α ) =
X
wij,α pi,α pj,α .
(7.15b)
i,j
The life cycle starts with zygotes in Hardy-Weinberg proportions. Selection acts in each
deme on the newly born offspring. Then recombination occurs followed by adult migration
and random mating within in each deme. This life cycle extends that in Section 3.2. To
deduce the general multilocus migration-selection dynamics (Nagylaki 2009b), let
x∗ij,α = pi,α pj,α wij,α /w̄α
(7.16a)
be the frequency of genotype ij in deme α after selection, and
p#
i,α =
X
Ri,j` x∗j`,α
(7.16b)
j,`
its frequency after recombination. Here, Ri,j` is the probability that during gametogenesis,
paternal haplotypes j and ` produce a gamete i by recombination. Finally, migration
occurs and yields the gamete frequencies in the next generation in each deme:
p0i,α =
X
mαβ p#
i,β .
(7.16c)
β
The recurrence equations (7.16) describe the evolution of gamete frequencies under selection on multiple recombining loci and migration. We view (7.16) as a dynamical system
on SΓI . We leave it to the reader to check the obvious fact that the processes of migration
and recombination commute.
The complications introduced by recombination are disguised by the terms Ri,jl which
depend on the recombination frequencies among all subsets of loci. To obtain an analytically useful representation of (7.16b), more effort is required.
Let {K, N} be a nontrivial decomposition of L, i.e., K and its complement N = L \ K are
each proper subsets of L and, therefore, contain at least one locus. (The decompositions
66
{K, N} and {N, K} are identified.) We designate by cK the probability of reassociation
of the genes at the loci in K, inherited from one parent, with the genes at the loci in N,
inherited from the other. Let
X
ctot =
cK ,
(7.17)
K
where
P
K
runs over all (different) decompositions {K, N} of L, denote the total recombi-
nation frequency. We designate the recombination frequency between loci k and n, such
that k < n, by ckn . It is given by
ckn =
X
cK ,
(7.18)
K∈Lkn
where Lkn = {K : k ∈ K and n ∈ N} (Bürger 2000, p. 55). Unless stated otherwise, we
assume that all pairwise recombination rates ckn are positive. Hence,
cmin = min ckn > 0 ,
k<n
(7.19)
We define
Di,α =
1 XX
cK (wij,α pi,α pj,α − wiK jN ,jK iN ,α piK jN ,α pjK iN ,α ) .
w̄α j K
(7.20)
This is a measure of LD in gamete i in deme α. A considerable generalization of the
arguments leading to (7.3) shows that (7.16b) can be expressed in the following form
(Nagylaki 1993):
wi,α
− Di,α .
(7.16b’)
w̄α
If there is only one deme, then (7.16b’) provides the recurrence equation describing evop#
i,α = pi,α
lution under selection on multiple recombining loci. It seems worth noting that the full
dynamics, (7.16), does not depend on linkage disequilibria between demes.
Let
n
o
(1)
(L)
Λ0,α = p·,α : pi,α = pi1 ,α · . . . · piL ,α ⊆ SI
(7.21)
denote the linkage-equilibrium manifold in deme α, and let
Λ0 = Λ0,1 × . . . × Λ0,Γ ⊆ SΓI .
(7.22)
If there is no position effect, i.e., if wij,α = wiK jN ,jK iN ;α for every i, j, and K, then Di,α = 0
for every p·,α ∈ Λ0,α . Hence,
Λ0,α ⊆ {p·,α : D·,α = 0} ,
67
(7.23)
where D·,α is defined in analogy to (7.12b). In the absence of selection, equality holds in
(7.23).
We will often need the following assumption:
The backward migration matrix M is ergodic, i.e., irreducible and aperiodic.
(E)
Given irreducibility, the biologically trivial condition that individuals have positive probability of remaining in some deme, i.e., mαα > 0 for some α, suffices for aperiodicity (Feller
1968, p. 426). Because M is a finite matrix, ergodicity is equivalent to primitivity.
If (E) holds, there exists a principal left eigenvector ξ ∈ intSΓ such that
ξ>M = ξ> .
(7.24)
The corresponding principal eigenvalue 1 of M is simple and exceeds every other eigenvalue
in modulus. The principal eigenvector ξ is the unique stationary distribution of the
Markov chain with transition matrix M , and M n converges at a geometric rate to eξ > as
n → ∞, where e = (1, . . . , 1)> ∈ RΓ (Feller 1968, p. 393; Seneta 1981, p. 9).
We average pi,α with respect to ξ,
P i = ξ > pi ,
P = (P1 , . . . , PI )> ∈ SI ,
(7.25)
and define the gamete-frequency deviations q from the average gamete frequency P :
qi,α = pi,α − Pi ,
(7.26a)
qi = pi − Pi e ∈ RΓ ,
X
(k)
qi ∈ RΓ ,
qik =
(7.26b)
q·,α = p·,α − P ∈ RI ,
(7.26d)
i|ik
> >
>
q = (q·,1
, . . . , q·,Γ
) ∈ RIΓ .
(7.26c)
(7.26e)
Therefore, q measures spatial heterogeneity or diversity. If q = 0, the gametic distribution
is spatially homogeneous.
7.4
Migration and recombination
We study migration and recombination in the absence of selection, i.e., if wij,α = 1 for
every i, j, α. Then the dynamics (7.16) reduces to
X
p0i,α =
mαβ (pi,β − Di,β ) ,
β
68
(7.27)
where
Di,β =
X
(K) (N)
cK pi,β − piK ,β piN ,β
(7.28)
K
and {K, N} as above (7.17). Here,
(K)
piK ,β =
X
pi,β ,
(7.29)
i|iK
P
where i|iK runs over all multi-indices i with the components in K fixed as iK , denotes
the marginal frequency of the gamete with components ik fixed for the loci k ∈ K. In
vector form, i.e., with the notation (7.12a), (7.27) becomes
p0i = M (pi − Di ) ,
i ∈ I.
(7.30)
The following theorem shows that in the absence of selection trajectories quickly approach LE and spatial homogeneity.
Theorem 7.1 (Bürger 2009a). Suppose that (7.30) and (E) hold. Then, the manifold
Y (k)
(k)
Ψ0 = p ∈ SΓI : pi =
pik and qik = 0 for every k and ik
(7.31a)
k
= p∈
SΓI
: D = 0 and q = 0
(7.31b)
is invariant under (7.30) and globally attracting at a uniform geometric rate. Furthermore,
every point on Ψ0 is an equilibrium point. Thus, (global) LE and spatial homogeneity are
approached quickly under recombination and (ergodic) migration.
This theorem generalizes the well known fact that in multilocus systems in which
recombination is the only evolutionary force, linkage disequilibria decay to zero at a
geometric rate (Geiringer 1944, Lyubich 1992). The proof is very technical and uses
induction on the number of embedded loci of the full L-locus system. Theorem 7.1 does
not hold if the migration matrix is reducible or periodic (Remark 3.3 in Bürger 2009a).
In the following, we investigate several limiting cases in which useful general results
can be proved and perturbation theory allows important extensions.
7.5
Weak selection
We assume that selection is weaker than recombination and migration. The main result
will be that all trajectories converge to an invariant manifold Ψ close to Ψ0 (7.31), on
which there is LE and allele frequencies are deme independent. On Ψ , the dynamics
69
can be described by a small perturbation of a gradient system on Ψ0 . This implies that
all trajectories converge, i.e., no cycling can occur, and the equilibrium structure can be
inferred from that of the much simpler gradient system.
Throughout this section, we assume (E), i.e., the backward migration matrix is ergodic.
We assume that there are constants rij,α ∈ R such that
wij,α = 1 + rij,α ,
(7.32)
where ≥ 0 is sufficiently small and |rij | ≤ 1. Migration and recombination rates, mαβ
and cK , are fixed so that fitness differences are small compared with them. From (7.15)
and (7.32), we deduce
wi,α (p·,α ) = 1 + ri,α (p·,α ) ,
w̄α (p·,α ) = 1 + r̄α (p·,α ) ,
(7.33)
in which
ri,α (p·,α ) =
X
rij,α pj,α ,
r̄α (p·,α ) =
j
X
rij,α pi,α pj,α .
(7.34)
i,j
When selection is dominated by migration and recombination, we expect that linkage
disequilibria within demes as well as gamete- and gene-frequency differences between
demes decay rapidly to small quantities. In particular, we expect approximately panmictic
evolution of suitably averaged gamete frequencies in ‘quasi-linkage equilibrium’. We also
show that all trajectories converge to an equilibrium point, i.e., no complicated dynamics,
such as cycling, can occur. In the absence of migration, this was proved by Nagylaki et al.
(1999, Theorem 3.1). For a single locus under selection and strong migration, this is the
content of Theorem 4.5 in Nagylaki and Lou (2007). Theorem 7.2 and its proof combine
and extend these results as well as some of the underlying ideas and methods.
To formulate and prove this theorem, we define the vector
T
(1)
(1)
(L)
(L)
ρα = p1,α , . . . , pI1 ,α , . . . , p1,α , . . . , pIL ,α ∈ SI1 × · · · × SIL
(7.35)
of all allele frequencies at every locus in deme α, and the vector
T
(1)
(1)
(L)
(L)
π = P1 , . . . , PI1 , . . . , P1 , . . . , PIL
∈ SI1 × · · · × SIL
(7.36)
of all averaged allele frequencies at every locus. We note that in the presence of selection
(k)
the Pik , hence π, are time dependent. Instead of p, we will use π, D, and q to analyze
(7.16), and occasionally write p = (π, D, q).
70
On the LE manifold Λ0,α (7.21), which is characterized by the ρα (α ∈ G), the selection
coefficients of gamete i, allele in at locus n, and of the entire population are
ri,α (ρα ) =
X
rij,α
Y
j
(7.37a)
k
rin ,α (ρα ) =
X
r̄α (ρα ) =
X
(n)
(k)
pjk ,α ,
Y
ri,α (ρα )
(k)
pik ,α ,
(7.37b)
(k)
(7.37c)
k:k6=n
i|in
ri,α (ρα )
Y
i
pik ,
k
cf. (7.34). As in (7.24), let ξ denote the principal left eigenvector of M . We introduce
the average selection coefficients of genotype ij, gamete i, allele in at locus n, and of the
entire population:
ωij =
X
ωi (π) =
X
ξα rij,α ,
(7.38a)
α
ωij
Y
j
(n)
ωin (π)
=
X
ωi (π)
Y
(k)
Pi k
=
X
(7.38b)
ωi (π)
Y
i
ξα rin ,α (π) ,
(n)
(7.38c)
ξα r̄α (π) .
(7.38d)
α
k6=n
X
ξα ri,α (π) ,
α
k
X
i|in
ω̄(π) =
(k)
Pj k =
(k)
P ik =
X
α
k
For ω̄, we obtain the alternative representations
!
ω̄(π) =
XX
n
(n)
(n)
ωin Pin =
X
ωij
Y
n
i,j
in
and
dω̄(π)
(n)
dPin
(n)
Pi n
(n)
= 2ωin (π) .
!
Y
(k)
Pj k
,
(7.38e)
k
(7.39)
For reasons that will be justified by the following theorem, we call the differential
equation
(n)
(n)
Ṗin = Pin
h
i
(n)
ωin (π) − ω̄(π) ,
D = 0, q = 0
(7.40a)
(7.40b)
on SΓI the weak-selection limit of (7.16). In view of the following theorem, it is more
convenient to consider (7.40a) and (7.40b) on SΓI instead of (7.40a) on SI1 × · · · × SIL .
The differential equation (7.40a) is a Svirezhev-Shashahani gradient (Remark 2.3) with
71
potential function ω̄. In particular, ω̄ increases strictly along nonconstant solutions of
(7.40a) because
ω̄˙ = 2
XX
n
(n)
P in
h
i2
(n)
ωin (π) − ω̄(π) ≥ 0 .
(7.41)
in
We will also need the assumption
All equilibria of (7.40a) are hyperbolic.
(H)
Theorem 7.2 (Bürger 2009a). Suppose that (7.16), (7.32), (E) and (H) hold, the backward migration matrix M and all recombination rates cK are fixed, and > 0 is sufficiently
small.
(a) The set of equilibria Ξ0 ⊂ SΓI of (7.40) contains only isolated points, as does the
set of equilibria Ξ ⊂ SΓI of (7.16). As → 0, each equilibrium in Ξ converges to the
corresponding equilibrium in Ξ0 .
(b) In the neighborhood of each equilibrium in Ξ0 , there exists exactly one equilibrium
point in Ξ . The stability of each equilibrium in Ξ is the same as that of the corresponding
equilibrium in Ξ0 ; i.e., each pair is either asymptotically stable or unstable.
(c) Every solution p(t) of (7.16) converges to one of the equilibrium points in Ξ .
The essence of this theorem and its proof is that under weak selection (i) the exact
dynamics quickly leads to spatial quasi-homogeneity and quasi-linkage equilibrium, and
(ii) after this time, the exact dynamics can be perceived as a perturbation of the weakselection limit (7.40). The latter is much easier to study because it is formally equivalent
to a panmictic one-locus selection dynamics. Theorem 7.2 is a singular perturbation result
because in the absence of selection every point on Ψ0 is an equilibrium.
Parts (a) and (b) of the above theorem follow essentially from Theorem 4.4 of Karlin
and McGregor (1972b) which is an application of the implicit function theorem and the
Hartman-Grobman theorem. Part (c) is much stronger and relies, among others, on the
notion of chain-recurrent points and their properties under perturbations of the dynamics
(see Section 6.3).
Outline of the proof of Theorem 7.2. Theorem 7.1 and the theory of normally hyperbolic
manifolds imply that for sufficiently small s, there exists a smooth invariant manifold Ψ
close to Ψ0 , and Ψ is globally attracting for (7.16) at a geometric rate (see Nagylaki et
al. 1999, and the references there). The manifold Ψ is characterized by an equation of
the form
(D, q) = ψ(π, ) ,
72
(7.42)
where ψ is a smooth function of π. Thus, on Ψ , and more generally, for any initial values,
after a long time,
D(t) = O() and q(t) = O() .
(7.43)
The next step consists in deriving the recurrence equations in an O() neighborhood
of Ψ0 which, in particular, contains Ψ . By applying (7.32) and D(t) = O() to (7.16b’),
straightforward calculations yield
(n) #
pin ,α
=
(n)
pin ,α
+
(n)
(n) rin ,α (ρα ) − r̄α (ρα )
pin ,α
w̄α (ρα )
+ O(2 )
(7.44)
for every α ∈ G (see eq. (3.10) in Nagylaki et al. 1999). By averaging, invoking (7.16),
(7.44), and (7.42) one obtains
(n) 0
Pi n
(n) 0
(n) #
(n) #
= µT p in = µT M p in = µT pin
h
i
(n)
(n)
(n)
= Pin + Pin ωin (π) − ω̄(π) + O(2 ) .
(7.45)
(n) 0
By rescalinghtime as τ = t
i and letting → 0, the leading term in (7.45), Pin =
(n)
(n)
(n)
Pin + Pin ωin (π) − ω̄(π) , approximates the gradient system (7.40a). Therefore, we
have ω̄(π 0 ) > ω̄(π) unless π 0 = π. In particular, the dynamics (7.45) on Ψ0 is gradient
like.
The eigenvalues of the Jacobian of (7.45) have the form 1 + ν + O(2 ), where ν
signifies an eigenvalue of the Jacobian of (7.40a). Therefore, (H) implies that also every
equilibrium of (7.45) is hyperbolic.
The rest of the proof is identical to that of Theorem 3.1 in Nagylaki et al. (1999). The
heart of its proof is the following. Since solutions of (7.16) are in phase with solutions on
the invariant manifold Ψ , it is sufficient to prove convergence of trajectories for initial
conditions p ∈ Ψ . With the help of the qualitative theory of numerical approximations,
it can be concluded that the chain-recurrent set of the exact dynamics (7.16) on Ψ is a
small perturbation of the chain-recurrent set of (7.40a). The latter dynamics, however,
is gradient like. Because all equilibria are hyperbolic by assumption, the chain-recurrent
set consists exactly of those equilibria. Therefore, the same applies to the chain-recurrent
set of (7.16) if > 0 is small.
A final point to check is that unstable boundary equilibria remain in the simplex
after perturbation. The reason is that an equilibrium p̂ of (7.40) on the boundary of
SΓI satisfies p̂i = 0 for every i in some subset of I, and this condition is not altered by
migration. The positive components of p̂ are perturbed within the boundary, where the
equilibrium remains.
73
Our next result concerns the average of the (exact) mean fitnesses over demes:
X
w̄(p) =
ξα w̄α (p·,α ) .
(7.46)
α
Theorem 7.3 (Bürger 2009a). Suppose the assumptions of Theorem 7.2 apply. If (7.43)
holds, π is bounded away from the equilibria of (7.40a), and p is within O(2 ) of Ψ , then
∆w̄(p) > 0 .
The time to reach an O(2 ) neighborhood of Ψ can be estimated (Remark 4.6 in
Bürger 2009a and Remark 4.13 in Nagylaki and Lou 2007). Unless migration is very
weak or linkage very tight, convergence occurs in an evolutionary short period. It follows
that mean fitness is increasing during the long period when most allele-frequency change
occurs, i.e., after reaching Ψ and before reaching a small neighborhood of the stable
equilibrium.
An essential step in the proof is to show that
(D0 , q 0 ) − (D, q) = O(2 )
(7.47)
is satisfied if (7.43) holds. Therefore, linkage disequilibria and the measure q of spatial
diversity change very slowly on Ψ . This justifies to call states on Ψ spatially quasihomogeneous and to be in quasi-linkage equilibrium.
Theorem 7.2 enables the derivation of the following result about the equilibrium structure of the multilocus migration-selection dynamics (7.16). It establishes that arbitrarily
many loci with arbitrarily many alleles can be maintained polymorphic by migrationselection balance:
Theorem 7.4. Let L ≥ 1, Γ ≥ 2, In ≥ 2 for every n ∈ L, and let all recombination rates
cK be positive and fixed.
(a) There exists an open set Q of migration and selection parameters, such that for
every parameter combination in Q, there is a unique, internal, asymptotically stable equilibrium point. This equilibrium is spatially quasi-homogeneous, is in quasi-linkage equilibrium, and attracts all trajectories with internal initial condition. Furthermore, every
trajectory converges to an equilibrium point as t → ∞.
(b) Such an open set, Q0 , also exists if the set of all fitnesses is restricted to be
nonepistatic and to display intermediate dominance.
For a more detailed formulation and a proof, see Theorem 2.2 in Bürger (2009b). The
constructive proof suggests that with increasing number of alleles and loci, the proportion
74
of parameter space that allows for a fully polymorphic equilibrium shrinks rapidly because
at every locus some form of overdominance of spatially averaged fitnesses is required. In
addition, the proof shows that this set Q exists in the parameter region where migration
is strong relative to selection.
Remark 7.5. The set Q0 in the above theorem does not contain parameter combinations
such that all one-locus fitnesses are multiplicative or additive, or more generally satisfy
DIDID (5.26). In each of these cases, one gamete becomes fixed as t → ∞ (Proposition
2.6 and Remark 2.7 in Bürger 2009b).
7.6
Strong migration
Now we assume that selection and recombination are weak relative to migration, i.e., in
addition to (7.32), we posit
cK = γK ,
K ⊆ L,
(7.48)
where ≥ 0 is sufficiently small and γK is defined by this relation. Then every solution
p(t) of the full dynamics (7.16) converges to a manifold close to q = 0. On this manifold,
the dynamics is approximated by the differential equation
i
X h
(K) (N)
γK Pi − PiK PiN ,
Ṗi = Pi [ωi (P ) − ω̄(P )] −
(7.49a)
K
which we augment with
q = 0.
(7.49b)
This is called the strong-migration limit of (7.16).
In general, it cannot be expected that the asymptotic behavior of solutions of (7.16)
under strong migration is governed by (7.49) because its chain-recurrent set does not always consist of finitely many hyperbolic equilibria. The differential equation (7.49a) is the
continuous-time version of the discrete-time dynamics (7.16b’) which describes evolution
in a panmictic population subject to multilocus selection and recombination. Although
the recombination term is much simpler than in the corresponding difference equation,
the dynamics is not necessarily less complex. Akin (1979, 1982) proved that (7.49) may
exhibit stable cycling. Therefore, under strong migration and if selection and recombination are about equally weak, convergence of trajectories of (7.16) will not generally occur,
and only local perturbation results can be derived (Bürger 2009a, Proposition 4.10).
The dynamics (7.49) becomes simple if there is a single locus (then the recombination
term in (7.49a) vanishes) or if there is no epistasis, i.e., if fitnesses can be written in the
75
form
ωij =
X
(n)
uin jn
(7.50)
n
for constants
(n)
u in jn
≥ 0. Essentially, this means that fitnesses can be assigned to each
one-locus genotype and there is no nonlinear interaction between loci. Then mean fitness
is a Lyapunov function, all equilibria are in LE, and global convergence of trajectories
occurs generically (Ewens 1969, Lyubich 1992, Nagylaki et al. 1999).
7.7
Weak recombination
If recombination is weak relative to migration and selection, the limiting dynamics is
formally equivalent to a multiallelic one-locus migration-selection model, as treated in
Sections 3 – 6. Then local perturbation results can be applied but, in general, global
convergence of trajectories cannot be concluded and, in fact, does not always occur.
7.8
Weak migration
In contrast to the one-locus case treat in in Section 6.3, in the multilocus case the assumption of weak migration is insufficient to guarantee convergence of trajectories. The
reason is that in the absence of migration the dynamics (7.16) reduces to
p0i,α = p#
i,α = pi,α
wi,α
− Di,α
w̄α
(7.51)
for every i ∈ I and every α ∈ G; cf. (7.16b’). Therefore, we have Γ decoupled multilocus
selection dynamics, one for each deme. Because already for a single deme, stable cycling
has been established (Hastings 1981, Hofbauer and Iooss 1984), global perturbation results can not be achieved without additional assumptions. Such an assumption is weak
epistasis.
(n)
We say there is weak epistasis if we can assign (constant) fitness components uin jn ,α > 0
to single-locus genotypes, such that
wij,α =
X
(n)
uin jn ,α + ηsij,α ,
(7.52)
n
where the numbers sij,α satisfy |sij,α | ≤ 1 and η ≥ 0 is a measure of the strength of
epistasis. It is always assumed that η is small enough, so that wij,α > 0.
For the rest of this section, we assume weak migration, i.e., (6.19) and (6.20), weak
epistasis (7.52), and
η = η() ,
76
(7.53)
where η : [0, 1) → [0, ∞) is C 1 and satisfies η(0) = 0. Therefore, migration and epistasis
need not be ‘equally’ weak. In particular, no epistasis (η ≡ 0) is included.
In the absence of epistasis and migration ( = η = 0), p is an equilibrium point of
(7.51) if and only if for every α ∈ G, p·,α is both a selection equilibrium for each locus and
is in LE (Lyubich 1992, Nagylaki et al. 1999). In addition, the only chain-recurrent points
of (7.51) are its equilibria (Lemmas 2.1 and 2.2 in Nagylaki et al. 1999). Therefore, one
can apply the proof of Theorem 2.3 in Nagylaki et al. (1999) to deduce the following result
which simultaneously generalizes Theorem 2.3 in Nagylaki et al. (1999) and Theorem 6.5:
Theorem 7.6 (Bürger 2009a, Theorem 5.4). Suppose that in the absence of epistasis
every equilibrium of (7.51) is hyperbolic, and > 0 is sufficiently small.
(a) The set of equilibria Σ0 ⊂ SΓI of (6.21) contains only isolated points, as does the
set of equilibria Σ ⊂ SΓI of (7.16). As → 0, each equilibrium in Σ converges to the
corresponding equilibrium in Σ0 .
(b) In the neighborhood of each asymptotically stable equilibrium in Σ0 , there exists
exactly one equilibrium in Σ , and it is asymptotically stable. In the neighborhood of each
unstable internal equilibrium in Σ0 , there exists exactly one equilibrium in Σ , and it is
unstable. In the neighborhood of each unstable boundary equilibrium in Σ0 , there exists at
most one equilibrium in Σ , and if it exists, it is unstable.
(c) Every solution p(t) of (7.16) converges to one of the equilibrium points in Σ .
In contrast to the case of weak selection (Theorem 7.2), unstable boundary equilibria
can leave the state space under weak migration (Karlin and McGregor 1972a). For an
explicit example in the one-locus setting, see Remark 4.2 if Nagylaki and Lou (2007).
If = 0, then all equilibria are in Λ0 , i.e., there is LE within each deme, but not
between demes. If > 0 is sufficiently small, then there is weak LD within each deme,
i.e., Di,α = O() for every i and every α.
Remark 7.7. If in Theorem 7.6 the assumption of weak epistasis is not made, statements
(a) and (b) remain valid, but (c) does not necessarily follow. Statement (c) follows if, for
= 0, the chain recurrent set consists precisely of the hyperbolic equilibria.
One of the consequences of the above theorem is the following, which contrasts sharply
with Theorem 7.4:
Theorem 7.8 (Bürger 2009b, Theorem 3.9). For an arbitrary number of loci, sufficiently
weak migration and epistasis, and partial dominance, the number of demes is the generic
77
maximum for the number of alleles that can be maintained at any locus at any equilibrium
(stable or not) of (7.16).
7.9
Weak evolutionary forces
If all evolutionary forces are weak, i.e., if → 0 in (7.32), (6.19), and
cK = γK
for all subsets K ⊆ L ,
(7.54)
the limiting dynamics of (7.16) on SΓI becomes
X
X (K) (N)
ṗi,α = pi,α [ri,α (p·,α ) − r̄α (p·,α )] −
γK pi,α − piK ,α piN ,α +
µαβ pi,β .
K
(7.55)
β
Hence, selection, recombination, and migration are decoupled. This dynamics may be
viewed as the continuous-time version of (7.16). However, it may exhibit complex dynamical behavior already if either migration or recombination is absent. Analogs of Theorems
7.1, 7.2, 7.3, 7.4, and 7.6 apply to (7.55).
7.10
The Levene model
For the Levene model, the recurrence equations (7.16) simplify to
X
p0i =
Ri,j` pj p` cα wj`,α /w̄α , i ∈ I ,
(7.56)
j,`,α
where this form is sufficient to deduce the main results, i.e., it is not necessary to express
the recombination probabilities Ri,j` in terms of the linkage disequilibria.
Many of the results on the one-locus Levene model in Sections 5.1 and 5.2 can be generalized to the multilocus Levene model if epistasis is absent or weak. These generalizations
are based on two key results. The first is that in the absence of epistasis geometric mean
fitness is again a Lyapunov function, and the internal equilibria are its stationary points
(Theorems 3.2 and 3.3 in Nagylaki 2009b). The second key result is that in the absence
of epistasis, generically, every trajectory converges to an equilibrium point that is in LE
(Theorem 3.1 in Bürger 2010). This is true for the haploid and the diploid model. Then a
proof analogous to that of Theorem 7.6 yields a global perturbation result for weak epistasis, in particular, generic global convergence to an equilibrium point in quasi-linkage
equilibrium (Theorem 7.2 in Bürger 2010).
With the aid of these results, the next two theorems can be derived about the maintenance of multilocus polymorphism in the Levene model.
78
Theorem 7.9 (Bürger 2010, Result 5.1). Assume an arbitrary number of multiallelic
loci, Γ ≥ 2 demes of given size, and let all recombination rates be positive and fixed.
Then there exists a nonempty open set of fitness parameters, exhibiting partial dominance
between every pair of alleles at every locus, such that for every parameter combination in
this set, there is a unique, internal, asymptotically stable equilibrium point of the dynamics
(7.56). This equilibrium is globally asymptotically stable.
Theorem 7.9 is the multilocus extension of Theorem 5.8. From a perturbation theorem
for weak epistasis in the Levene model, analogous to Theorem 7.6 (Theorem 7.2 in Bürger
2010), it follows that such an open set exists also within the set of non-epistatic fitnesses.
The following result is of very different nature and generalizes Proposition 3.18 in
Nagylaki (2009b):
Theorem 7.10 (Bürger 2010, Theorem 7.4). Assume weak epistasis and diallelic loci
with DIDID.
(a) If L ≥ Γ, at most Γ − 1 loci can be maintained polymorphic for an open set of
parameters.
(b) If L ≤ Γ − 1 and if the dominance coefficients in (5.26) are arbitrary for each locus
but fixed, there exists an open nonempty set W of fitness schemes satisfying (7.52) and of
deme proportions (c1 , . . . , cΓ−1 ), such that for every parameter combination in W, there
is a unique, internal, asymptotically stable equilibrium point of (7.56). This equilibrium
is in quasi-linkage equilibrium and globally attracting. The set W is independent of the
choice of the recombination rates.
Whereas Theorem 7.9 suggests that spatially varying selection has the potential to
maintain considerable multilocus polymorphism, Theorem 7.10 shows that properties of
the genetic architecture of the trait (no dominance or DIDID) may greatly constrain this
possibility. Numerical results in Bürger (2009c) quantify how frequent polymorphism is
in the two-locus two-allele Levene model, and how this depends on the selection scheme
and dominance relations. With increasing number of loci or number of alleles per locus,
the volume of parameter space in which a fully polymorphic equilibrium is maintained
will certainly decrease rapidly.
A few other aspects of the multilocus Levene model have also been investigated. Zhivotovsky et al. (1996) employed a multilocus Levene model to study the evolution of phenotypic plasticity. Wiehe and Slatkin (1998) explored a haploid Levene model in which
LD is caused by epistasis. More recently, van Doorn and Dieckmann (2006) performed
79
a numerical study that admits new mutations (of variable effect) and substitutions at
many loci. They showed that genetic variation becomes increasingly concentrated on a
few loci. Roze and Rousset (2008) derived recursions for the allele frequencies and for
various types of genetic associations in a multilocus infinite-island model. Barton (2010)
studied certain issues related to speciation using a generalized haploid multilocus Levene
model that admits habitat preferences.
7.11
Some conclusions
Local adaptation of subpopulations to their specific environment occurs only if the alleles
or genotypes that perform well are maintained within the population. Similarly, differentiation between subpopulations occurs only if they differ in allele or gamete frequencies.
Therefore, maintenance of polymorphism is indispensable for evolving or maintaining local adaptation and differentiation. The more loci and alleles are involved, the higher is the
potential for local adaptation and differentation. This is the main reason, why conditions
for the maintenance of polymorphism played such an important role in this survey.
Because for a single randomly mating population, polymorphic equilibria do not exist
in the absence of epistasis and of overdominance or underdominance, it is natural to confine attention to the investigation of the maintenance of genetic variation in subdivided
populations if in each subpopulation epistasis is absent (or weak) and dominance is intermediate. In addition, intermediate dominance seems to be by far the most common form
of dominance.
We briefly recall the most relevant results concerning maintenance of genetic variation
and put them into perspective. Particularly relevant results for a single locus are provided in Sections 4.2 – 4.5, and by Theorems 5.9, 5.16, 5.19, 6.1, and 6.9. Theorems 7.4,
7.8, 7.9, and 7.10 treat multiple loci. Theorems 7.4 and 7.9 establish that in structured
populations and in the Levene model, respectively, spatially varying selection can maintain multiallelic polymorphism at arbitrarily many loci under conditions for which in a
panmictic population no polymorphism at all can be maintained.
Interestingly, for strong migration and only two demes, an arbitrary number of alleles
can be maintained at each of arbitrarily many loci, whereas for weak migration, the
number of demes is a generic upper bound for the number of alleles that can be maintained
(Theorem 7.8). At first, this appears counterintuitive given the widespread opinion that
it is easier to maintain polymorphism under weak migration than under strong migration.
This opinion derives from the fact that for a single diallelic locus and two demes, the
80
parameter region for which a protected polymorphism exists increases with decreasing
migration rate, provided there is a single parameter that measures the strength of selection, as is the case in the Deakin model; see condition (4.19). Additional support for
this opinion comes from the study of weak migration in homogeneous and heterogeneous
environments (Karlin and McGregor 1972a,b; Christiansen 1999), from the analysis of the
general Deakin (1966) model (Section 4.3), as well as from various numerical studies (e.g.,
Spichtig and Kawecki 2004, Star et al. 2007a,b). However, Karlin (1982, p. 128) observed
that for the non-homogeneous Deakin model, in which the ‘homing probabilities’ vary
among demes, “it is possible to increase a single homing rate and reduce or even abrogate the event of A-protection”. This is already obvious from the protection condition
(4.14): If only one of the migration rates in (4.13) is reduced, it depends on the sign of
the corresponding selection coefficient if protection is facilitated or not.
Theorem 7.4 is not at variance with Theorem 7.8 or the widespread opinion expressed
above. It is complimentary, and its proof yields deeper insight into the conditions under
which genetic variation can be maintained by migration-selection balance. With strong
migration, a stable multiallelic polymorphism requires some form of overdominance of
suitably averaged fitnesses for the loci maintained polymorphic. The reason is that strong
migration leads to strong mixing, so that gamete and allele frequencies become similar
among demes. Because the fraction of volume of parameter space, in which average
overdominance holds for multiple alleles, shrinks rapidly with increasing number of alleles
or loci, we expect very stringent constraints on selection and dominance coefficients for
maintaining polymorphism at many loci if migration is strong.
The conditions for maintaining loci polymorphic under weak migration are much
weaker. With arbitrary intermediate dominance, the proof of Theorem 7.8 shows that
those alleles will be maintained that are the fittest in at least one niche. This, obviously,
limits the number of alleles that can be maintained.
Also in the Levene model arbitrarily many loci can be maintained polymorphic in the
absence of epistasis and if dominance is intermediate in every deme (Theorem 7.9). But
there, the maximum number of polymorphic loci depends on the pattern of dominance
and the number of demes (Nagylaki 2009b, Bürger 2010, and Theorem 7.10). It is an open
problem if for every (ergodic) migration scheme, arbitrarily many loci can be maintained
polymorphic in the absence of epistasis, overdominance, and underdominance.
Although the fraction of parameter space, in which the conditions for maintenance
of multiallelic multilocus polymorphisms are satisfied, decreases rapidly, stable multiallelic polymorphism involving small or moderate numbers of alleles or loci does not seem
81
unlikely. What is required if dispersal is weak, basically, is that there is a mosaic of directional selection pressures and different genes or genotypes that are locally well adapted.
8
Application: A two-locus model for the evolution
of genetic incompatibilities with gene flow
Here we show how to apply some of the above perturbation results in conjunction with
Lyapunov functions and an index theorem to derive the complete equilibrium and stability
structure for a two-locus continent-island model, in which epistatic selection may be
strong. The model was developed and studied by Bank et al. (2012) to explore how much
gene flow is needed to inhibit speciation by the accumulation of so-called DobzhanskyMuller incompatibilities (DMI).
The idea underlying the concept of DMIs is the following. If a population splits into
two, for instance because at least one of the subpopulations moves to a different habitat, and the two subpopulations remain separated for a sufficiently long time, in each of
them new, locally adapted alleles may emerge that substitute previous alleles. Usually,
such substitutions will occur at different loci. For instance, if ab is the original haplotype
(gamete), also called wild type, in one population Ab may become fixed (or reach high
frequency), whereas in the other population aB may become fixed. If individuals from the
derived populations mate, then hybrids of type AB, which occur by recombination between aB and Ab have sometimes reduced fitness, i.e., they are incompatible (to a certain
degree). Therefore, the accumulation of DMIs is considered a plausible mechanism for the
evolution of so-called intrinsic postzygotic isolation between allopatric (i.e., geographically
separated) populations. Since complete separation is an extreme situation, Bank et al.
(2012) studied how much gene flow can inhibit the accumulation of such incompatibilities,
hence parapatric speciation.
As a simple scenario, Bank et al. (2012) assumed a continent-island model, i.e., a
continental population, which is fixed for one haplotype (here, aB), sends migrants to
an island population in which, before the onset of gene flow, Ab was most frequent or
fixed. To examine how much gene flow is needed to annihilate differentiation between the
two populations, the equilibrium and stability structure was derived. A stable internal
equilibrium, at which all four haplotypes are maintained, is called a DMI because, only
when it exists, is differentiation between the two populations maintained at both loci.
Bank et al. investigated more general biological scenarios than the one outlined above, and
they investigated a haploid and a diploid version of their model. Here, we deal only with
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the haploid case, as it admits a much more complete analysis and is also representative
for an important subset of diploid models.
Our goal is to derive the equilibrium and bifurcation structure of the haploid model.
We convey the main ideas and only outline most of the proofs. Detailed proofs are given in
the Online Supplement S1 of Bank et al. (2012). Also comprehensive biological motivation
and discussion is provided by Bank et al., as well as illuminating figures.
8.1
The haploid DMI model
Let x1 , x2 , x3 , and x4 denote the frequencies of the four haplotypes ab, aB, Ab, and AB
P
on the island. They satisfy xi ≥ 0 for every i and 4i=1 xi = 1. We assume that selection
acts on individuals during the haploid phase of their life cycle according to the following
scheme for Malthusian fitness values:
haplotype
fitness
frequencies
wild type continental
ab
aB
w1 = 0
x1
w2 = β
x2
island
Ab
recombinant
AB
w3 = α
x3
w4 = α + β − γ
x4
Table 8.1: Haplotype fitnesses and frequencies in the DMI model
This scheme is entirely general. The fitness of the wild type is arbitrarily normalized
to 0. The parameters α and β measure a potential selective advantage of the island and
continental haplotypes, respectively, on the island. They can be positive or negative.
However, further below we will assume that α > 0 because, as will be shown, otherwise
a DMI cannot exist. Finally, the epistasis parameter γ measures the strength of the
incompatibility among the A and B alleles. We assume γ ≥ 0, so that epistasis is negative
or absent (γ = 0).
Assuming weak evolutionary forces in continuous time, the haplotype dynamics is given
by
ẋi = xi (wi − w̄) − ηi rD − mxi + δ2i m,
(8.1)
where r is the recombination rate between the two loci, D = x1 x4 − x2 x3 is the LD,
η1 = η4 = −η2 = −η3 , and δi2 = 1 if i = 2 and δi2 = 0 otherwise; cf. (7.10) and (7.55).
83
With the settings as in Table 8.1, we obtain
ẋ1 = x1 [−α(x3 + x4 ) − β(x2 + x4 ) + γx4 ] − rD − mx1 ,
(8.2a)
ẋ2 = x2 [−α(x3 + x4 ) + β(x1 + x3 ) + γx4 ] + rD + m(1 − x2 ) ,
(8.2b)
ẋ3 = x3 [α(x1 + x2 ) − β(x2 + x4 ) + γx4 ] + rD − mx3 ,
(8.2c)
ẋ4 = x4 [α(x1 + x2 ) + β(x1 + x3 ) − γ(x1 + x2 + x3 )] − rD − mx4 .
(8.2d)
This is a dynamical system on the simplex S4 which constitutes our state space. We
always assume m ≥ 0, r ≥ 0, and γ ≥ 0.
For many purposes, it is convenient to describe the dynamics in terms of the allele
frequencies p = x3 + x4 of A, q = x2 + x4 of B, and the measure D of LD. Then the
dynamical equations read
ṗ = αp(1 − p) − γ(1 − p)(pq + D) + βD − mp ,
(8.3a)
q̇ = βq(1 − q) − γ(1 − q)(pq + D) + αD + m(1 − q) ,
(8.3b)
Ḋ = [α(1 − 2p) + β(1 − 2q)]D − γ[(1 − p)(1 − q) − D](pq + D)
− rD − m[p(1 − q) + D] .
(8.3c)
It is straightforward to show that the condition x ∈ S4 translates to 0 ≤ p ≤ 0, 0 ≤ q ≤ 0,
and
− min[pq, (1 − p)(1 − q)] ≤ D ≤ min[p(1 − q), (1 − p)q] .
8.2
(8.4)
Existence and stability of boundary equilibria
We denote the monomorphic equilibria xi = 1 by Mi . If m = 0, then all monomorphic
equilibria exist. However, if α > 0 and γ > β, the conditions most relevant for this
investigation (see (8.17) below), M1 and M4 are always unstable.
It is straightforward to determine local stability of M2 and M3 . The latter can be
stable only if m = 0.
Next, there may exist two equilibria at which one locus is polymorphic and the other is
m
, 1, 0 and is admissible
fixed. The equilibrium SA has the coordinates (p, q, D) = 1 − α−γ
if and only if m < α − γ. The equilibrium SB has coordinates 0, − m
,
0
and is admissible
β
if and only if m < −β. The respective local stability conditions are easy to derive.
The following lemma collects the important observations from these analyses.
84
Lemma 8.1. (a) For given m > 0, at most one of the boundary equilibria M2 , SA , or
SB can be stable. M2 is asymptotically stable if
m > max[−β, α − γ, α − β − r] .
(8.5)
SA is asymptotically stable if
γ <α+β
and
(α − γ)(γ − β)
α+β−γ
1+
<m<α−γ,
α
r
which requires r > γ − β. SB is asymptotically stable if
−βα
γ−β−α
γ > α + β and
1+
< m < −β ,
γ−β
r
(8.6)
(8.7)
which requires r > α.
(b) If M2 is asymptotically stable, then SA and SB are not admissible.
(c) As a function of m, boundary equilibria change stability at most once. If a change
in stability occurs, then it is from unstable to stable (as m increases).
Finally, if r = 0, there is a fully polymorphic equilibrium R0 on the edge x1 = x4 = 0
of S4 . Thus, only the island and the continental haplotypes are present. Its coordinates
are easily calculated.
In the following we derive global asymptotic stability of boundary equilibria for various
sets of parameters by applying the theory of Lyapunov functions (e.g. LaSalle 1976, in
particular, Theorem 6.4 and Corollary 6.5). By global asymptotic stability of an equilibrium we mean that every trajectory, such that initially all alleles are present, converges
to this equilibrium. By Lemma 8.1 there is at most one asymptotically stable boundary
equilibrium for any given set of parameters. Hence, convergence of all trajectories to
the boundary is sufficient for demonstrating global stability. Because global convergence
to the boundary precludes the existence of an internal equilibrium, it yields necessary
conditions for a stable DMI.
Proposition 8.2. A DMI can exist only if each of the following conditions is satisfied:
α > 0 and γ > β ,
(8.8)
m < α−β +r,
(8.9)
m < max{α − β, 14 α, 41 (γ − β)} .
(8.10)
85
Condition (8.8) means that w(Ab) > max{w(ab), w(aB)}, i.e., the island type has
higher fitness than its one-step mutational neighbors.
The proof of (8.8) uses the Lyapunov functions
Y =
x1 + x3
1−q
=
x3 + x4
p
(8.11)
X=
x1 + x3
1−q
=
x1 + x2
1−p
(8.12)
and
to show that convergence to SA or M2 occurs if (8.8) is violated.
Condition (8.9) follows because, if (8.8) is satisfied,
ẋ2 = x2 [β(x1 + x3 ) + γx4 − α(x3 + x4 ) − rx3 ] + m(x1 + x3 + x4 ) + rx1 x4
≥ x2 [(m + β)x1 + (m + β − α − r)x3 + (m − α + γ)x4 ]
(8.13)
holds if
m > max{−β, α − γ, α − β + r} = α − β + r .
(8.14)
Therefore, global convergence to M2 occurs if (8.14) holds. In particular, M2 is globally
asymptotically stable for every m if r < β − α.
Condition (8.10) follows in a similar way.
8.3
Properties of internal equilibria
Lemma 8.3. Every trajectory eventually enters the region D ≤ 0 and remains there.
Convergence to D = 0 occurs if and only if at least one allele is eventually lost. Thus,
every internal equilibrium satisfies D < 0.
To prove this lemma, we define
Z=
x2 x 3
,
x1 x4
(8.15)
where x1 > 0 and x4 > 0 is assumed. We note that Z = 1 if and only if D = 0, and Z < 1
if and only if D > 0. Then
Ż = x1 x3 x4 (m + γx2 ) + rD(x1 x2 x3 + x2 x3 x4 + x1 x2 x4 + x1 x3 x4 ) .
(8.16)
We observe that Ż ≥ 0 holds whenever D ≥ 0. In addition, it follows immediately that
Ż > 0 if rD > 0 and x2 + x3 > 0. If x2 + x3 = 0 and x1 x4 > 0, then ẋ1 + ẋ4 < 0 if
r + m > 0. Hence, all trajectories leave D > 0 if r > 0. If rD = 0, then Ż = 0 only if
86
x3 = 0 or if m = 0 and γx2 = 0. Thus, our result follows by investigating (i) the dynamics
on x3 = 0 if r = 0, (ii) the dynamics on x2 = 0 if m = r = 0, and (iii) the case m = γ = 0.
We leave the simple first two cases to the reader. The third case is also not difficult and
follows immediately from Section 3.4.1 in Bürger and Akerman (2011).
From here on, we assume
α > 0 and γ > β and r > β − α ,
(8.17)
because we have proved that internal equilibria can exist only if (8.17) is satisfied. We
note that (8.17) holds if and only if M3 (island haplotype fixed) is linearly stable in the
absence of migration.
Our next aim is the derivation of a cubic equation from which the coordinate p of an
internal equilibrium (p, q, D) can be obtained. Given p, the coordinates q and D can be
computed from relatively simple explicit formulas.
By solving ṗ = 0, we find that, for given p and q, and if p 6= 1 − β/γ, the value of LD
at equilibrium is
m + (1 − p)(γq − α)
.
(8.18)
β − γ + γp
Substituting this into (8.3b), assuming β 6= 0, and solving q̇ = 0 for q, we obtain
p
1
m
q1,2 (p) =
1−
± Q ,
(8.19)
2
β
D = D(p, q) = p
where
2
m
4αmp
4α(γ − α)
Q= 1+
−
−
p(1 − p)
(8.20)
β
β(γ − β)
β(γ − β)
needs to be nonnegative to yield an admissible equilibrium. Finally, we substitute q =
q1 (p) and D = D(p, q1 (p)) into (8.3c) and obtain that an equilibrium value p must solve
the equation
(γ − β)A(p) −
p
QB(p) = 0 ,
(8.21)
where
A(p) = (γ − β) β[(α − r)(γ − 2α) + γ(α − β)] + [β(γ + β − 2α) + r(2β − γ)]m + βm2
+ β [2α(2γ − 3β)(γ − α) − βγ(γ − β)] − (2γ − β)(2α − γ)r
+ [2αβ(γ − 2β) − βγ(γ − β) + γ(2γ − 3β)r]m p
− [2αβ(γ − α)(γ − 2β) + βγ(γ − 2α)r + γ 2 rm]p2 ,
(8.22a)
B(p) = (γ − β)[−2αβ + γ(β + r) + βm]
+ [2αβ(γ − β) − βγ(γ − β) + r(β − 2γ)]p + γ 2 rp2 .
87
(8.22b)
If we substitute q = q2 (p) and D = D(p, q2 (p)) into (8.3c), we obtain
p
(γ − β)A(p) + QB(p) = 0 ,
(8.23)
instead of (8.21).
By simple additional considerations, it is shown that solutions p of (8.21) or (8.23)
satisfy
0 = (γ − β)2 A(p)2 − QB(p)2
4β
=
(β − γ + γp)2 [m + (γ − α)(1 − p)]P (p) ,
γ−β
(8.24)
where
P (p) = (γ − β)(m + r + β − α)[αβ(α + β − γ − r) − mr(γ − β)]
+ αβ(α − β)(γ − β)(α + β − γ) + α(γ − β)[3β(γ − 2α) + m(4β − γ)]r
+ [α(γ 2 + βγ − β 2 ) + γ(γ − β)m]r2 p
− 2αr[β(γ − β)(γ − 2α) + γ 2 r]p2 + αγ 2 r2 p3 .
(8.25)
Because p = 1 − β/γ never gives an equilibrium of (8.3) and p = 1 − m/(γ − α) can give
rise only to a single-locus polymorphism, any internal equilibrium value p must satisfy
P (p) = 0.
We can summarize these findings as follows.
Theorem 8.4. (a) The haploid dynamics (8.3) can have at most three internal equilibria,
and the coordinate p̂ of an internal equilibrium (p̂, q̂, D̂) is a zero of the polynomial P given
by (8.25).
(b) For given p̂ with P (p̂) = 0, only one of q1 (p̂) or q2 (p̂) defined in (8.19) can yield
an equilibrium value q̂. D̂ is calculated from p̂ and q̂ by (8.18). This procedure yields an
internal equilibrium if and only if 0 < p̂ < 1, 0 < q̂ < 1, and
− min[p̂q̂, (1 − p̂)(1 − q̂)] < D̂ < 0 .
(8.26)
Remark 8.5. In the absence of epistasis, i.e., if γ = 0, there can be at most two internal equilibria. Their coordinates are obtained from a quadratic equation in p. The
admissibility conditions are given by simple formulas (Bürger and Akerman 2011).
If r = 0, the equilibrium R0 mentioned in Section 8.2 is obtained from the unique zero
of P (p).
For a number of important special or limiting cases, explicit expressions or approximations can be obtained for the internal equilibria.
88
8.4
Bifurcations of two internal equilibria
A bifurcation of two internal equilibria can occur if and only if P (p∗ ) = 0, where p∗ ∈ (0, 1)
is a critical point of P , i.e., P 0 (p∗ ) = 0. There are at most two such critical points, and
they are given by
√ o
1 n
2
p∗1,2 =
2α[β(γ
−
2α)(γ
−
β)
+
γ
r]
±
R ,
(8.27a)
3αγ 2 r
where
R = α2 −β(γ − β) 16αβ(γ − α)(γ − β) + γ 2 (3α2 + β 2 ) − γ 3 (3α + β)
− βγ 2 (γ − β)(γ − 2α)r + γ 2 (3β 2 − 3βγ + γ 2 )r2
− 3mrαγ 2 (γ − β)[4αβ + γ(r − α)].
(8.27b)
Solving either P (p∗1 ) = 0 or P (p∗2 ) = 0 for m, we obtain after some straightforward manipulations that the critical value m∗ must be a solution of the following quartic equation:
[αβ(2β −γ)(α+β −γ +r)+γ(γ −β)rm]2 [−ψ1 +2ψ2 rm+27αγ 2 (γ −β)r2 m2 ] = 0 , (8.28a)
where
ψ1 = α(γ − β)(α − β + r)2 [4αβ(γ − α)(γ − β) + γ 2 r2 ] ,
(8.28b)
ψ2 = 2α2 (γ − β)2 (9βγ − 8αβ − αγ) + 3αγ(γ − β)(3βγ − 2αβ − αγ)r
+ 3αγ 2 (γ − β)r2 + 2γ 3 r3 .
(8.28c)
The zero m = m0 arising from the first (linear) factor in (8.28a) does not give a valid
bifurcation point for internal equilibria.
The second (quadratic) factor in (8.28a) provides two potential solutions. However,
because ψ1 ≥ 0, one is negative. Therefore, the critical value we are looking for is given
by
q
1
2
2
m =
−ψ2 + ψ2 + 27αγ (γ − β)ψ1 .
(8.29)
27αγ 2 (γ − β)r
At this value, two equilibria with non-zero allele frequencies collide and annihilate each
∗
other. Thus, m∗ is the critical value at which a saddle-node bifurcation occurs. This
gives an admissible bifurcation if both equilibria are internal (hence admissible) for either
m < m∗ or m > m∗ .
If p∗1 = p∗2 , i.e., if R = 0, a pitchfork bifurcation could occur at m∗ . As a function of
α, β, and γ, the condition p∗1 = p∗2 can be satisfied at m∗ only for three different values
89
of r, of which at most two can be positive. It can be shown that at each of these values,
one of the emerging zeros of P (p) does not give rise to an admissible equilibrium (because
D > 0 there). Thus, only a saddle-node bifurcation can occur.
Simple and instructive series expansions for m∗ may be found in the Online Supplement
of Bank et al. (2012).
8.5
No migration
We assume m = 0. From Section 8.3, we obtain the following properties of internal
equilibria (p, q, D). The LD is given by
D = D(p, q) = p(1 − p)
γq − α
,
β − γ + γp
(8.30)
where p 6= 1 − β/γ; cf. (8.18). For admissibility, (8.26) needs to be satisfied. For given
p and if β 6= 0, the coordinate q of an internal equilibrium can assume only one of the
following forms:
1
q1,2 (p) =
2
s
1±
4α(γ − α)p(1 − p)
1−
β(γ − β)
!
.
(8.31)
By Theorem 8.4, for given p, at most one of q1 = q1 (p) or q2 = q2 (p) can give rise to an
equilibrium.
The following theorem characterizes the equilibrium and stability structure.
Theorem 8.6. Suppose (8.17) and m = 0.
(a) The haploid dynamics (8.3) admits at most one internal equilibrium.
(b) Depending on the parameters, the internal equilibrium is given by either
(p, q1 (p), D(p, q1 (p))) or (p, q2 (p), D(p, q2 (p))), where p is one of the two zeros of
P (p) = γ 2 r2 p2 − r[2β(γ − β)(γ − 2α) + rγ 2 ]p + β(γ − β)(α − β − r)(α + β − γ − r) , (8.32)
and qi (p) and D(p, qi (p)) are given by (8.31) and (8.30), respectively.
(c) An internal equilibrium exists if and only if both M2 and M3 are asymptotically
stable. This is the case if and only if
γ > α and β > 0 and r > α − β .
(d) The internal equilibrium is unstable whenever it exists.
(e) If (8.33) does not hold, then M3 is globally asymptotically stable.
90
(8.33)
This theorem complements the results derived by Feldman (1971) and Rutschman
(1994) for the discrete-time dynamics of the haploid two-locus selection model. Rutschman
proved global convergence to a boundary equilibrium for all parameter combinations for
which no internal equilibrium exists. If transformed to the parameters used by Rutschman,
condition (8.33) yields precisely the cases not covered by Rutschman’s Theorem 14. Because our model is formulated in continuous time, the internal equilibrium can be determined by solving quadratic equations. This is instrumental for our proof, as is the
following index theorem.
Remark 8.7 (Hofbauer’s index theorem). Theorem 2 in Hofbauer (1990) states the following. For every dissipative semiflow on Rn+ such that all fixed points are regular, the
sum of the indices of all saturated equilibria equals +1.
In our model, the index of an equilibrium is (−1)m , where m is the number of negative
eigenvalues (they are always real). An internal equilibrium is always saturated. If it
is asymptotically stable, it has index 1. Equilibria on the boundary of the simplex are
saturated if and only if they are externally stable. This is the case if and only if no gamete
that is missing at the equilibrium can invade. Because S4 is attracting within R4+ , the
index of an asymptotically stable (hence, saturated) boundary equilibrium is 1.
Outline of the proof of Theorem 8.6. Statements (a) and (b) follow from four technical
lemmas that provide properties of the polynomial P (p) and resulting admissibility conditions for (p, q, D).
The proof of (c) and (d) is the mathematically most interesting part. In view of the
above, it remains to prove that the internal equilibrium exists if (8.33) holds and that it
is unstable. This follows readily from the index theorem of Hofbauer (1990) in Remark
8.7. In our model, the only boundary equilibria are the four monomorphic states. M1 and
M4 are never saturated because they are unstable within S4 . M2 and M3 are saturated
if and only if they are asymptotically stable within S4 . Then, ind(M2 ) = ind(M3 ) = 1.
Hence there must exist an internal equilibrium with index -1. Such an equilibrium cannot
be stable. Because M2 and M3 are both asymptotically stable if and only if (8.33) holds,
statements (c) and (d) are proved.
(e) follows with the help of several simple Lyapunov functions.
8.6
Weak migration
Theorem 7.6 and Remark 7.7 in conjunction with Theorem 8.6 and Lemma 8.1 yield the
equilibrium configuration for weak migration.
91
Theorem 8.8. If m > 0 is sufficiently small, the following equilibrium configurations
occur.
(a) If (8.33) holds, there exists one unstable internal equilibrium and one asymptotically stable internal equilibrium (the perturbation of M3 ). In addition, the monomorphic
equilibrium M2 is asymptotically stable. Neither SA nor SB is admissible.
(b) Otherwise, i.e., if γ < α or β < 0 or r < α − β, the perturbation of the equilibrium
M3 is globally asymptotically stable (at least if γ is small). The equilibrium M2 is unstable,
and the equilibria SA and SB may be admissible. If SA or SB is admissible, it is unstable.
Throughout, we denote the stable internal equilibrium by IDMI . To first order in m its
coordinates are
8.7
m(α + r)
m(γ − β + r)
m
1−
,
,−
α(α − β + r) (γ − β)(α − β + r) α − β + r
.
(8.34)
The complete equilibrium and stability structure
Now we are in the position to prove our main results about the equilibrium and bifurcation
structure. We continue to assume (8.17), so that a DMI can occur. Throughout, we always
consider bifurcations as a function of (increasing) m.
We define
(α − γ)(γ − β)
α+β−γ
mA =
1+
,
α
r
−βα
γ−β−α
mB =
1+
,
γ−β
r
m2 = α − β − r
and note that mA , mB , and m2 are the critical values of m above which SA , SB ,
respectively, are asymptotically stable provided they are admissible. If we set

m
if γ − α < 0 < γ − β < α and r > γ − β,

 A
m−
mB
if β < 0 < α < γ − β and r > α,
max =


m2
if r ≤ min[α, α − β, γ − β],
(8.35a)
(8.35b)
(8.35c)
and M2 ,
(8.36a)
(8.36b)
(8.36c)
then, by Lemma 8.1, all boundary equilibria are repelling if and only if m < m−
max . As
the following theorem shows, if m < m−
max , an asymptotically stable internal equilibrium
exists which, presumably, is globally attracting. Hence, a DMI will evolve from every
initial condition. Of course, m−
max can be zero (if 0 < α < β < γ and r ≤ β − α). We also
recall the definition of m∗ from (8.29).
92
Theorem 8.9. The following three types of bifurcation patterns can occur:
Type 1.
• If 0 < m < m∗ , there exist two internal equilibria; one is asymptotically stable
(IDMI ), the other (I0 ) is unstable. The monomorphic equilibrium M2 is asymptotically stable.
• At m = m∗ , the two internal equilibria collide and annihilate each other by a saddlenode bifurcation.
• If m > m∗ , M2 is the only equilibrium; it is asymptotically stable and, presumably,
globally stable.
Type 2. There exists a critical migration rate m̃ satisfying 0 < m̃ < m∗ such that:
• If 0 < m < m̃, there is a unique internal equilibrium (IDMI ). It is asymptotically
stable and, presumably, globally stable.
• At m = m̃, an unstable equilibrium (I0 ) enters the state space by an exchange-ofstability bifurcation with a boundary equilibrium.
• If m̃ < m < m∗ , there are two internal equilibria, one asymptotically stable (IDMI ),
the other unstable (I0 ), and one of the boundary equilibria is asymptotically stable.
• At m = m∗ , the two internal equilibria merge and annihilate each other by a saddlenode bifurcation.
• If m > m∗ , a boundary equilibrium asymptotically stable and, presumably, globally
stable.
Type 3.
• If 0 < m < m−
max , a unique internal equilibrium (IDMI ) exists. It is asymptotically
stable and, presumably, globally stable.
• At m = m−
max , IDMI leaves the state space through a boundary equilibrium by an
exchange-of-stability bifurcation.
• If m > m−
max , a boundary equilibrium is asymptotically stable and, presumably,
globally stable.
Proof. Theorem 8.8 provides all equilibrium configurations for small m. Lemma 8.1 provides control over the boundary equilibria. As m increases, they can vanish but not
emerge. They can also become asymptotically stable as m increases. For sufficiently large
m, there is always a globally asymptotically stable boundary equilibrium. By Theorem
93
a)
IDMI
M2
m
c)
b)
IDMI
IDMI
SA or SB
I0
I0
SA or SB
M2
M2
m
m
e)
d)
IDMI
IDMI
SA or SB
SA or SB
M2
M2
m
m
Figure 8.1: Bifurcation patterns in the DMI model according to Theorem 8.9. Panel a shows
the bifurcation pattern of Type 1. Panels b and c display the two cases that are of Type 2, and
panels d and e those of Type 3.
94
8.4, the number of internal equilibria is at most three. In addition, internal equilibria
can emerge or vanish only either by a saddle-node bifurcation (Section 8.4) or because
an equilibrium enters or leaves S4 through one of boundary equilibria, when an exchange
of stability occurs. A bifurcation involving the two internal equilibria can occur at most
at one value of m, namely at m∗ (8.29). An exchange-of-stability bifurcation can occur
only at the values mA , mB , or m2 . If it occurs, the respective boundary equilibrium is
asymptotically stable for every larger m for which it is admissible. By the index theorem
of Hofbauer (1990), Remark 8.7, the sum of the indices of all saturated equilibria is 1.
If Case 1 of Theorem 8.8 applies, then M2 is asymptotically stable for every m > 0, and
it is the only boundary equilibrium. Hence, its index is 1. The index of the stable internal
equilibrium is also 1. Because the sum of the indices of the internal equilibria must be 0,
the index of the unstable equilibrium is -1. Because at most one bifurcation involving the
two internal equilibria can occur and because for large m, M2 is globally asymptotically
stable, the bifurcation must be of saddle-node type in which the equilibria collide and
annihilate each other (but do not emerge). In principle, the internal equilibria could
also leave S4 through a boundary equilibrium (in this case, it must be M2 ). However,
by the index theorem, they can do so only simultaneously. This occurs if and only
if m∗ = m2 , which is a non-generic degenerate case. Because the sum of indices of
the internal equilibria must be zero, no equilibrium can enter the state space. These
considerations settle the bifurcation pattern of Type 1.
If Case 2 of Theorem 8.8 applies, then, for small m, the boundary equilibria are unstable, hence not saturated, and do not contribute to the sum of indices. Since then the
indices of internal equilibria must sum up to 1, the only possible bifurcation that does
not entail the stability of a boundary equilibrium would be a pitch-fork bifurcation of the
internal equilibrium which, by Section 8.4, does not occur. Indeed, because m∗ is the only
value at which a bifurcation among internal equilibria can occur, and because for large m
a boundary equilibrium is globally asymptotically stable, the three equilibria emerging by
a pitchfork bifurcation would have to leave the state space through boundary equilibria.
This, however, cannot occur, as follows easily from the results about linear stability in
Section 8.2. Thus, any further bifurcations involve a boundary equilibrium. There are
two possibilities.
(i) An equilibrium enters S4 at some value m̃ (which can only be one of mA , mB , or m2 )
through one of the unstable boundary equilibria by an exchange-of-stability bifurcation.
If m > m̃, there is one asymptotically stable boundary equilibrium, an unstable internal
equilibrium (the one that entered S4 ), and one asymptotically stable internal equilibrium.
95
Now a reasoning analogous to that applied above to Case 1 of Theorem 8.8 establishes
the bifurcation pattern of Type 2.
(ii) The internal equilibrium leaves S4 by exchange of stability through one of the
boundary equilibria at m−
max . This becomes asymptotically stable then and, presumably,
globally stable. At larger values of m no equilibrium can enter S4 through one of the
(other) unstable boundary equilibria, because this would either lead to two simultaneously
stable boundary equilibria, which is impossible (Lemma 8.1), or this had to occur at the
same value at which the hitherto stable boundary equilibrium merges with M2 and leaves
the state space. This, too, is impossible because the sum of the indices of the new stable
boundary equilibrium and the new unstable internal equilibrium would be zero. Thus, we
have established the bifurcation pattern of Type 3 and excluded all other possibilities.
Our final goal is to assign the respective parameter combinations to the three types of
bifurcation patterns determined above. To this aim we define four selection scenarios that
reflect different biological situations. We say that the fitness landscape is of slope type if
at least one of the one-mutant neighbors (ab or AB) has fitness intermediate between the
continental (aB) and the island type (Ab). Otherwise, there is a double-peak landscape.
Selection scenario 1 : 0 < α < β < γ. It represents the parameter regime in which “selection against hybrids” (of continental and island haplotypes) is driving DMI evolution.
The fitness landscape has two peaks with the higher at the continental type. The genetic
incompatibility is strong (γ large).
Selection scenario 2 : 0 < β < α < γ. It represents an intermediate parameter
regime in which the two mechanisms of “selection against hybrids” and “selection against
immigrants” (below) interfere. This is also a double-peak landscape, but with maximum
at the island type. The incompatibility is strong.
Selection scenario 3 : β < γ < min{α, α + β}. It represents part of the parameter regime in which “selection against immigrants” drives DMI evolution. The fitness
landscape is of slope type and the incompatibility is weak. β may be positive or negative.
Selection scenario 4 : β < 0 < α < γ − β. It represents part of the parameter regime
in which “selection against immigrants” drives DMI evolution. The fitness landscape is
of slope type and there is strong local adaptation (β < 0 < α).
96
To formulate the main result, we need the following critical recombination rates:
3(γ − β) − α
,
2γ − α
3α + β − γ
rB = β
,
β+γ
p
3α(γ − β) − α(γ − β)(4βγ + 5αγ − 9αβ)
.
r2 =
2γ
rA = (γ − α)
Theorem 8.10. 1. Bifurcation patterns of Type 1 occur in
Selection scenario 1;
Selection scenario 2 if and only if r ≥ α − β.
2. Bifurcation patterns of Type 2 occur in
Selection scenario 2 if and only if r2 < r < α − β;
Selection scenario 3 if and only if one of the following holds:
(a) r2∗ < r ≤ γ − β,
(b) r > max[γ − β, rA ] and γ > 21 α,
(c) γ − β < r < ∞ and γ = 21 α > 3β,
(d) γ − β < r < rA and γ < 21 α.
Selection scenario 4 if and only if one of the following holds:
(a) r2∗ < r ≤ α,
(b) r > max[α, rB ] and γ > −β,
(c) α < r < ∞ and γ = −β < 3α + β,
(d) α < r < rB and γ < −β.
3. Bifurcation patterns of Type 3 occur in
Selection scenario 2 if and only if r ≤ r2 ;
Selection scenario 3 if and only if one of the following holds:
(a) r ≤ min[γ − β, r2∗ ],
(b) γ − β < r ≤ rA and γ > 21 α,
(c) γ − β < r < ∞ and γ = 21 α < 3β.
(d) r ≥ max[γ − β, rA ] and γ < 21 α.
Selection scenario 4 if and only if one of the following holds:
(a) r ≤ min[α, r2∗ ],
(b) α < r ≤ rB and γ > −β,
(c) α < r < ∞ and γ = −β ≥ 3α + β,
(d) r ≥ max[α, rB ] and γ < −β.
97
(8.37a)
(8.37b)
(8.37c)
For the quite tedious proof, we refer to the Online Supplement of Bank et al. (2012).
The theorem shows that for selection scenario 1, in which the continental type is
superior to all others, only the bifurcation pattern of Type 1 occurs (provided (8.17)
is assumed). Hence, if a DMI exists, it is only locally stable and initial conditions,
or historical contingencies, determine whether differentiation between the populations is
maintained or not. The only other situation in which the bifurcation pattern of Type
1 can occur is for the second double-peak scenario, provided linkage between the two
loci is sufficiently tight. For slope-type fitnesses (selection scenarios 3 and 4) as well
as for scenario 2 with strong recombination, the DMI is always globally stable for weak
migration.
If, in selection scenario 4, γ(α + β) < β(2α + β) and γ < −β hold, then a bifurcation
pattern of Type 3 occurs for every r > 0. These two conditions are satisfied whenever
−β > max[2α, γ]. Therefore, in the strong local adaptation scenario a globally asymptotically stable DMI occurs whenever the selection intensity on the two loci differs by more
than a factor of two. A bistable equilibrium pattern can occur in this scenario only if the
selection strength on both loci is sufficiently similar and the recombination rate is about
as strong as the selection intensity.
In summary, two mechanisms can drive the evolution of parapatric DMIs. In a heterogeneous environment, a DMI can emerge as a by-product of selection against maladpated
immigrants. In a homogeneous environment, selection against unfit hybrids can maintain a DMI. No DMI can be maintained if the migration rate exceeds one of the bounds
given in Proposition 8.2. Therefore, it is the adaptive advantage of single substitutions
rather than the strength of the incompatibility that is the most important factor for the
evolution of a DMI with gene flow. In particular, neutral DMIs can not persist in the
presence of gene flow. Interestingly, selection against immigrants is most effective if the
incompatibility loci are tightly linked, whereas selection against hybrids is most effective
if they are loosely linked. Therefore, opposite predictions result concerning the genetic
architecture that maximizes the rate of gene flow a DMI can sustain.
9
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